Functions Operations and Factoring

Okay, operations involving functions. Okay, so when we say operations, we are referring to the four fundamental operations that you have learned, right? And these are what? Addition, subtraction, multiplication, and division. Okay, but may dadagdag po tayong isa and they add the last one is the composition of functions. So that will be added in the operations involving functions. So let's have here different functions. Let's say function f of x is 3x to the 4th plus 2x minus 7. Then we have function g of x which is negative 5x3 minus 9x plus 12. We have function h of x which is 12x to the fourth minus 3x raised to 3 plus 5x minus 10. And then we have the k of x which is 7x to the second power plus 5x minus 11. So as you can see we have four different functions. So we're going to apply the operations but we start first in addition and subtraction. Then after that, we mix the addition and subtraction. So let's start from the basic. So let's do this one. kindly add function f to function h number two subtract function k to function g okay so with your okay uh previous knowledge about how to add and subtract polynomials can you do it on your own without discussing it first let me see if you can still remember how to add and subtract polynomials. These are examples of polynomials, right? Okay, so simply add. Paano daw tayo nag-a-add? Balikan po natin yung ating learning from Okay, balikan po natin yung learning daw natin saan? Sa rules of integers and sa ano pa po yun? Sa rules of integers at saka po sa exponent. How are we going to add if like sign? And if unlike sign, what happens to the exponent? On the board, you want it on the board. Okay, number one. Okay, on that side. And any volunteer naman po for number two? Anyone po for number two who wants to show his or her answer po sa board? Just give it a try. For number two po, who wants to find the sum and the difference naman po for number two? Number one po is sum because we are looking for the answer in addition. And number two naman is difference because we are looking for the answer in subtraction. Okay, let us check later. Okay, if that will be the answer. How about for example two? For example two po, who wants to give a try? Subtract function k to function g. No one? Okay, so let's us do first the number 1. Okay, so number 1. Okay, so we have here, okay, our function f which is 3x raised to 4 plus 2x minus 7 plus our function h which is 12x. to the fourth minus 3x3 plus 5x minus 10 okay so we have here okay so much better if we're going to that the terms in this uh in respective uh in respective group it is again combining like terms. So that is much better and more easier to see the sum of the function. Let's say we have here for our function f of x. So we have here 3x to the 4th 3x raised to 4th Since we don't have a raise to 3, right? So, let's just leave an additional space for x raised to 3. Then we have plus 2x and then minus 7. Then we add the g of x, add the h of x. So, that will be what is our h. So, 12x to the 4th, then minus 3x raised to the 3rd power plus 5x minus 3. As you can see, all the constants are in one line. Okay, all with x is in a similar line. Okay, and then we have x raised to 3 since we don't have x raised to 2. And then we have here our, those with x raised to 4 are also in a line. So now we have here, like sign, copy the sign, add the number. Unlike, like sign, copy the sign. Add the number, copy the variable, and then bring it down kasi wala naman siyang partner. So negative 3x raised to 3, and then add 15x raised to 4. Okay, now, let's go to the example 2. This time, okay, function k subtract to function g. The same process din po, but... What we're going to do when it comes to subtraction, we chase the sign of the second function and then proceed to addition then. Okay, example po, for number 2, we have function k, so we have 7x raised to 2 plus 5x minus 11. Subtract the function g. The function g automatically We will change the sign. Paltos subtraction. So from negative 5x raised to 3, it will be positive 5x raised to 3. Then, negative 9x will be positive 9x. And then, positive 12 will be negative 12. And then you add. Okay, like sign, copy the sign, add the number. Copy the sign, add the number copy the variable bring down since walang sign to ibi sabihin nun ay positive hindi po po pwedeng walang sign na and then we have 5x raised to 3 so this is the difference po for number two nasusundan po ba thumbs up po if you can follow okay isa pa this time kayo nagagawa ah Number three. Okay, now let's... have H plus K. And then number 4, in your place book, let's have, okay, F minus H. So for number 3, H plus K. And then for number 4, F minus H. So do it in your place. X, Y, raised to 4 minus 3x raised to 3 plus 5x minus 10 and then add down natin yung function k so 7x raised to 2 plus 5x and then minus 11 as you can see okay naka-align po yung mga similar terms so like sign copy the sign Add the number plus 10, copy the variable, plus 7x squared, baba ng po, and the last one. So that will be the sum. You got this answer. Okay, next po, for number 4. Okay? Pag ma-pinaba natin yung 7, you have to place po the plus sign. Kasi po, pag wala po yung plus sign, That will make your answer wrong. Kasi po, the number of terms po is depending on the plus and minus sign in between. Do you understand? So we should put a plus sign because it will make your answer wrong. Number 4 So function f So we have here 3x to the 4th plus 2x minus 7 Okay, so space 4 raised to 3, raised to 2 Then, since it is minus and h preceded by the minus sign Change the sign of all function h. We will change the sign automatic. So that will be negative 12x to the 4th, positive 3x raised to 3, negative 5x, and then plus 10. Then proceed to the addition. So now we have positive 3, because unlike sign, subtract, Copy the sign of the bigger number. Same with that. Next one. And then since we don't have x raised to 2, we will just add x raised to 3. We don't have an answer that has space. So we have 3x raised to 3. Then we have, unlike sign, so subtract. Copy the sign of 12. Negative 9x to the 4th. Who got this answer? Okay. Hindi po tayo pwede mag-answer ng negative 9x to the 4th plus 3x3 minus 3x plus 3. Nagkakaintindihan ba? Kailangan po ay it must be consecutive in descending order. Sunod-sunod po. Wala pong bakante. Nagkakaintindihan po ba? Okay, now. Who among you correctly answered number 3? Raise your hand, be honest. Or yung iba, hindi pa rin nasundan yung number 3? Okay, put down your hands. How about number 4? Who among you? Raise your hand. Okay. So now let's go to number 5. Let us give a try kapag tatlohan na. Okay. So, number 5. Sa lang muna. Let's say we have here K minus H. plus f. So k minus h plus f. Function k. So we have 7x raised to 2 plus 5x minus 11 minus function h. So ibig sabihin, we change the sign of all function h. So we have here negative 12 X to the 4th plus 3x3 minus 5x plus 10. So, we can do this first. So, we have here negative 1. Cancel out. If it's the same, like terms with different signs, cancellation. And then bring down 7x squared. Then we have positive 3x3 negative 12x to the 4th. Add do yung ating function f. So 3x to the 4th, positive 2x, and then minus 7. So we have now, this is positive 7x squared. So we have now negative 8, positive 2x, positive 7x squared. Positive 3x raised to 3 and negative 9x to the 4th. Okay, who got this answer? Okay. How about the others? Sanag kamali. So, pwede niyo pong gawin by 2's muna. Tapos, yung sagot doon, ituktong natin yung F. Para hindi kayo nagrito, kung tatlo siyang mayroong line. Kung hindi kayo sanay. Nakaka-intindihan ba? Okay, anyway, wala pong ano. Okay, in what way po, kung anong gusto nyo, which is easier for you. Wala pong problema as long as you have the solution and give the correct answers. Okay, let's have another one. Number six. Okay, so let's have here, this time. Okay, so we have... H, H, minus F, minus G. of x. h minus f minus g of x. Our function h, so that will be what? 12x raised to 4 minus 3 x raised to 3 plus 5x minus 10. Function f preceded by minus operation. So it will simply chase the sign of all function f. So that will be negative 3x to the 4th, negative 2x and positive 7. Then add po muna natin. Unlike sign, subtract, copy the sign of the bigger number. Unlike sign, subtract, copy the sign of the bigger number. Okay, wala. Then bring down, subtract 9x. to the fourth. Tapos, minus G. Yun G preceded by minus ole. So change the sign of all G. So that will be negative, a positive, 5x3, positive, 9x, and minus 12. Like sign, copy the sign, add the number. Copy the sign, add the number. Copy the variable. Since walang x raised to 2, it tatabi na. So positive 2x raised to 3, copy 9x to the 4th. So this will be the answer. You got this answer. Okay. So it seems na, okay, more than half po nakakakuha. So, okay, yung iba po ay... Better ask help na lang po on how. Yung mga hindi nakakasunod. Nakaka-incident ba? Okay, kasi hindi po pwede na. Paulit-ulit lang po tayo dito. Meron pa po tayong tatlong operations na i-discuss. Okay, next. Now we go to multiplying function. So, bawatan lang po natin una para hindi masyadong burden sa inyo. Okay, ayan po. Okay, ano naman daw yung ruling natin sa multiplying, diba? Okay, pag like sign, positive. Kapag unlike sign, negative. Multiply that number. Okay, and then add the exponent. Nakakaintindihan po ba? Okay, let me see. Okay, paano daw tayo mag-multiply this time on your own? It's like we're multiplying numbers. It's like 23 times 458. It's just like that. Okay, let's say we have here for number 1. Okay, function G times function F of X. Then number 2, we have function K times function H of X. Let's see. Number 1, so we have G. of x, so we have negative 5x raised to 3 minus 9x multiplied by function f, which is 3x to the 4th plus 2x minus 7. So, what we're going to do to make it easier, it's easier to multiply by vertical. Okay, so what we're going to do, this first term, We will multiply all of these once at a time. Can you understand? So we have sine first. Negative times positive is negative. Negative 5x3 times 3x to the 4th is what? Negative 15x7. You add the exponent. We are not multiplying the exponent. Can you understand? Okay, it means negative 5 times 3, negative 15. X3 times X4, add the exponent. Copy the variable, 3 plus 4 is 7. Isa-isa po. And then ito naman po, the next one. Negative 5X3 times 2X is what? Negative 10X to the 4th. Lagay natin dito, negative. 10x to the 4th Then ito uli Pangatlo Negative 5 x3 times negative 7, positive 35x raised to 3. So positive 35x raised to 3. Tapos ito naman, pagano naman din uli, isa-isa. Okay, negative 9x times 3x to the 4th. Negative 27x to the 5th. So negative... 27x to the 5th. Then, negative 9x times 2x. Negative 18x squared. So, negative 18x squared. Negative 9x times negative 7. Positive 63x. Then you add. So, positive 63x. Baba lang po natin. Baba lang po natin. Baba. Then baba. Then baba. And that's it. Who got this answer? Okay. Ilan lang po tumama? Dalawa. Okay. Sige. Gabi niyo ulit yung number 2. So now let's go to number 2. Again, K times H. So what is our K? 7x squared plus 5x Multiply daw sa ating h which is 12x to the 4th minus 3x3 plus 5x Again, ito muna daw na 7x squared Isa-isa dito sa function h So we have here So what is 7 times 12? 84. So 84x, what will be the exponent of 3x? 6. 2 plus 4, 6. Then, next one. 7x squared times negative 3x3. 7 times negative 3. Negative 21. Then x squared times x3, x raised to 5. Then, ito naman. 7x squared times 5x is what? Positive 35x raised to 3. Okay, who got the first line? Okay, now we go to the second line. This one, again. 5x times 12x to the 4th. 60x to the 3rd. Okay, positive 60x to the 5th. Ito, tsaka ito naman. So, 5x times negative 3x raised to 3. Negative 15x to the 4th. So, negative 15x to the 4th. 5x times 5x. So, positive 25x squared. And then you add. Walang ka partner, baba. Baba din. 35, x3, baba. Negative 15, x to the 4th, unlike sign, so you subtract. 16 minus 21? 39. What will be the sign? Positive. Positive. So positive, 39, x to the 5th, baba. 84, x raised to 6. Who got this answer? Okay, so next. What if we have the one? Last one. Okay. Yung iba, saan pa rin nagkakamali? Okay, sa exponent, anong ginagawa nyo sa exponent? Pinumultiply nyo? Oh, ina-add lang yan, diba? Kapag wala po siya exponent, that means imaginary one. Okay, next po. Isan try lang muna, f times h of x. f, so we have 3x to the 4th plus 2x minus 7 times do natin sa h, 12x to the 4th minus 3x3 plus 5x. So ito do muna, ito 3x raised to the 4th to all. Function of h. So one at a time. So 3x to the 4th times 12x to the 4th. What is what? 36x raised to 8. x raised to 8. 3x to the 4th times negative 3x raised to 3. Negative 9x. 9x raised to 7. 3x to the 4th times 5x. Positive 15x to the 5th. So positive. 15x to the 5th. Ito naman kong pangalawa na. Dito ulit sa lahat na to. One at a time. 2x times 12x to the 4th. 24x to the 5th. Dito. 24x to the 5th. 2x times negative 3x to the 3rd. Negative 6x. to the fourth then 2x times positive 5x positive 10x squared. We can add it first. So positive 10x squared minus 6x to the fourth positive 6 9x to the fifth minus 9x7 36x8. Let's go to the third one. Negative 7 times 12x to the 4th. Negative 84x to the 4th. Negative 84x to the 4th. Negative 7 times negative 3x raised to 3. Positive 21x raised to 3. So positive. So positive, what is this? 21. Where am I? 21x raised to 3. I just removed this. What is this? We just put it here. So positive 10x squared. Then we have, this again. Negative 7 times positive 5x, negative 35x. Then add. So we have negative 35x positive 10x squared. Saan? 30. Ay sorry po. 3, 9. So we have positive 21x3. Then like sign, negative 70x to the 4th, a 90. X to the 4th, and then positive 3, 9, X to the 5th, let's do it again, negative 9, X, 7, and 36, X, 8. There you go. You got this answer. Okay, although it's long, but if you're going to do it one at a time, you won't get confused. Do you understand? Minus F. What is our H? 16 X squared plus 11 X minus 19. Minus F. So change the sign of all turns of F. So we have negative 13 X to the fourth. Negative 9 X 3. And then positive 15 X. Then you add first. So we have negative 19, positive 26x, positive 16x squared, negative 9x3, and negative 13x to the 4th. Add down natin yung k. So we have positive 4x to the 4th, minus 13x squared, then we have 18. Then add. So negative 1, positive 26x, positive 13x squared, negative 9x3, and then negative 9x to the 4th. So this is the answer. Function G, negative 9x raised to 3 plus 11x. With minus 17x plus, so walang mapapalitang sign, 13x to the 4th plus 9. X3 minus 15x. So add. So minus, ano do to? 32x. Baba, positive 11x squared, cancelled. So 13x to the 4th. Tapos, minus H. Papagitan ng sign. So negative 16x. X squared, negative 11x, positive 19, add. So positive 19, copy the sign. So we have 43x. Then change the sign. Subtract the bigger number to the smaller number, copy the sign. So negative 5x squared. X squared, tabi natin to, 13x to the 4th. Okay, negative 9x raised to 3 plus 11x squared. Multiply by, okay, f, 13x to the 4th plus 9x3 minus 15x. Again, ito po muna. Negative 9 times 13, 13 times 9, 27, 117. So negative 117x to the 7th power. Then negative 9 times positive 9, so we have negative 81x to the 6th power. Then negative 9 times 15. So that will be positive 135x to the 4th. Okay, next. Dito naman tayo sa pangalawa. 11 times 13. Diba pag phone na mumultiply sa dito, pag iniwalayin lang. So that will be, okay, positive. One hundred forty-three. X to the sixth. Tapos eleven times nine. Is positive nine hundred, ah, ninety-nine. X to the fifth. Positive ninety-nine. X to the fifth. Eleven times nine. Two plus three, five. Then eleven times negative fifteen. Is negative 165x raised to 3. Okay, and then you add. So negative 165x raised to 3. Positive 134x to the 4th. Positive 99x to the 5th. Okay, magkaiba ang sign. Subtract. 1, 4, 3, minus 81, 2, 14, minus 8 is 6. So that will be positive 62x6. Then we have, okay, 117x7. Ayan po. We got it, 16x raised to 3. 2x times 13x, positive 26x raised to 2. Then we have negative 9 times negative 8, positive 72x raised to 2. Then negative 9 times 13, so we have 117. Okay, negative 117. Then add negative 117x. Add 19x squared below x3. I love you all. Let's say we have here. Let's say I have here. Example po ah. So we have x squared minus 9 over 3x plus. 9. Okay, now, can we simplify x squared minus 9 all over? over 3x plus 9. Can we do the cancellation here? Pwede po ba tayo mag-cancel? Okay, ano? How are we going to cancel this one? Pwede ba natin i-cancel yung 9 and 9 here? Oh. Okay, so we're going to get the factor first. So we cannot have, we cannot simplify this one. kasi po this is as one term kasi po we have the all over symbol or sign, nagkakaitin yan ba? okay kung pag ganito po sana siya x squared over 3x minus 9 over 9, yan po we can have the cancellation pero since ito po ay all over, ibig sabihin hindi po natin siya pwede i-separate, lagi po siya counted as you 1. Okay, remember po yung ating, let's say we have here 9 over 12. Diba? This is not yet the simplest form. So we can still simplify it by getting the factor of 9, which is 3 and 3 and then 4 and 3. 3 times 3 is 9 and then 4 times 3 is 12. So pag kinasal natin, that makes our 3-4. which is the simplest form of 9 over 12. Ang tawag natin doon is common factor. Okay, so gagawin natin, we're going to look for the common factor of the numerator and the denominator. So, isa-isa ni natin na, dito muna tayo sa binomial, hindi sabihin with two terms. Dalawa lang po yung pwede natin gamitin dyan. Okay, here, as you can see, Paano daw tawag natin dito sa x squared, x to the fourth, x to the sixth, x to the eighth? Ano daw tawag natin dito? Perfect variable. So pag sinabi natin perfect variable, it means that their exponents are even numbers. It can be divided into two. So what are the factors of x squared? And that is x times x. How about x to the 4th? x squared times x squared. How about x raised to 6? x 3 times x 3. How about x raised to 8? x raised to 4 times x raised to 4. Remember, kapag po like base, we simply copy the base and add the exponent. That's why we have copy the base x, 1 plus 1 is 2. Copy the base x, and then 2 plus 2 will be 4. This is what we call perfect variable. Ang tawag po natin dyan ay perfect variable. Now, this one naman is what we call perfect number or perfect square. So what are the perfect number or perfect square? So ito daw ay 1, 4, 9. 16, 25, 36, 49, and so on. What are the factors? Where did this 1 come from? 1 times 1. 2 times 2. 3 times 3. 4 times 4. 5 times 5. 6 times 6. What will be the perfect number after 36? 49. So that is 7 times 7. Okay, and then the operation is minus. What is the answer in subtraction? How do we call the answer in subtraction? Difference. So, ang tawag po dito ay difference of perfect square. Okay. So, ang tawag po dito is different of perfect square. If the first term is perfect number or variable, and the second term is a perfect number or variable, and the operation is minus. Okay. Another example we have here. Okay. Is this a perfect difference of perfect square when we have this one? What do you think? Is this under difference of perfect square? Yes, this is still under difference of perfect square. Why? Because 9 is... perfect number x raised to the 4 is a perfect variable the operation is minus and then the 25 is a perfect number. If this becomes plus, it is not under difference. We don't have the sum of the perfect square. Okay so if this is the given, it means what will be the factor of this one of number 1. That will be the square root of the first and second term and then two opposite signs. So, ganto po siya. X, X plus 3 minus 3. So, when we multiply this one, we will get X squared minus 9. So, here naman, paano daw yan? That will be another ulit. Root of 9, 3, 3. Root of X raised to 4, X squared. square x square one positive one negative root of 25 five are you following? okay let's have another one okay 16x6 minus 81y4 so what will be the factors of the following what is the factor? so that will be 16 4, 4, x3, minus 9, 9, and then y2, y2. When you multiply that, you will get 16x raised to 6, minus 81y raised to 4. Listen, so, if this is the given, Okay, this is not... A difference of perfect square. So this time, pag ganito naman, ang possible po niyan ay common factor. So ano daw yung common factor? What will be the common factor of our 3x plus 9? Okay, so the common factor is, okay, 3x plus 9, it means both terms is divisible by 3. So, factor out the 3. So, that is 3x divided by 3. The remaining is x. And 9 divided by 3 is 3. So, it means that this is 3 times x plus 3. And our sum is x plus 3 and x minus 3. Then, we do the cancellation. So now we can have x plus 3 over x plus 3, that is 1. So what's left in our numerator is x minus 3 and in our denominator will be 3. So that will be the simplest form. Nakakaintindihan po ba? That is when given the difference of the perfect square and a common factor. Nakakaintindihan ba? That is applicable kapag dalawang binomial ang given. Gawa lang po ako ng given. Plus 5x minus 14 over x squared minus 49. Okay, this time, we have the trinomial and binomial. Tandaan, kapag po binomial, there are two. possible lang po to find the factors. And that is if it is a difference of perfect square or there is a common factor. Okay, so dito sa baba, we all know na it is a difference of perfect square. Now, we go dito po sa upper part. So, hindi po natin ipakia-apply yung difference of perfect square kasi that is only applicable in binomial. Since it is trinomial, there is no... po silang common factor. Di po ba? So out yung dalawang yun. So as you can see, we can have factoring trinomial. So paano naman daw natin ipa-factor ang trinomial? So we have here x squared plus 5x minus 14. So yung ating pong last term, pupunin natin yung possible factors. Ibig sabihin, Numbers that when we multiply will produce 14. And that is 7 times 2 and 14 times 2. Now when you add or subtract, it will produce the middle term. So the possible is 7 and 2 because 7 minus 2 is 5. So 14 cannot produce the 5. So it means that the possible of Doña is 7 and 2. Then we have the root of X. word, so that will be x and x. Since it is minus, it means it's an opposite sign. Can you understand? Okay, then, since our middle term is positive, it means the one that is positive is the bigger number. 7 is bigger than 2, it means this is positive, this is negative. Always the bigger number, it always looks like Bye. yung sign ng middle term. Nakakaintindihan po ba? So, ito sabihin, when you multiply x plus 7 to x minus 2, the product is x squared plus 5x minus 14. So, ito daw yung factors niya. So, we have here x plus 7 and then x minus 2. Then, alam na natin yung factors ng baba. Ano doy factors ng baba? Okay, x you Plus 7 and x minus 7 because it is difference of perfect square. So we have now x plus 7 and x minus 7. Can we do the cancellation? Yes. Yes, here it is. So it's left at the top, x minus 2. It's left at the bottom, x minus 7. There's the answer. Can you understand? Okay, next. May isa pa po. Kumbaga, ano na yung nakuha natin? Okay, common factor, difference of perfect square, and factoring trinomials. Okay, next. Let's have another one. So, let's have 12x. Minus 4 over 3x squared plus 5x minus 2. Okay, so we have here 12x minus 4 over or divided by 3x squared plus 5x minus 2. So, what is the possible here? Common factor or difference of perfect square. Is this difference of perfect square? No. So, ang possible lang is common factor. Okay, now, can we apply what we did kamina dito? Can we apply yung makukuha natin daw yung middle term? By the factors of the last term? No. Because there is a coefficient beside x squared. Ang tawag naman po natin dito, If this is not there, we can use it as a factoring trinomial. If possible. Can you understand? But if there is a numerical coefficient here, what we call is the process of guess, check, and revise. It means, we will think of a number, we will fix it, we will check if it will be correct, if not, we will rearrange it again. We will check again if it will be correct. Tapos, rearrange ulit natin hanggang sa tumama. Ayun po yung guess, check, and revise. Okay. Huwag na po tayo pumunta ng quadratic formula. Ang tagal nun. Nagkakaintindihan ba? Okay. Now, let's go to this one. 3x squared plus 5x minus 2. So, dalawa ulit daw yung kanyang factors. Okay, 3 means 3 times 1. Wala nang iba, diba? So, we can have 3 here and 1 third. Tapos itong x, tag-isa. Hindi ko mo pwede na yung isang first term wala pong variable. Kailangan po, pareha silang may variable. Okay, so now, we have 2. So, dito maglalaro tayo sa 2. Ang 2 natin, it can be 2 times 1 lang. Wala nang iba. Okay, so let us place 2 here. and wonder then as you can see negative doto so it is a bn makaiba sila ng sign okay now let's Let's say we make it positive and we make it negative. Kasi sabi magka-opposite side. Hindi na po tayo mag-iisip na magpo-produce itong 5. Dapat mag-produce yung 5 kapag tinimes po natin itong ating inner sa outer. Tapos inad natin, lalabas dapat yung middle term. Okay. 2 times x. 2x. Positive 2x. 3x times negative 1, negative 3x. If we add it, that makes negative x. If we add this, 2x minus 3x, negative x. Did we get the middle? No. We changed the sign. Negative times positive, so it will be negative 2x, positive, positive x. Hindi pa rin. We change the number. 1 and 2. Paritan natin. So we have negative 1 times x. Negative x. 3x times 2. 6x. Negative x plus 6x when we add. Positive 5x. Check. Nagkakaintigyan ba? So hindi sabihin. Ang factor ng 3x squared plus 5x minus 2 is 3x minus 1 and x plus 2. That is what we call guess, check, and revise. Okay, so now, nakuha na natin ito. Ang factor doon ito ay 3x minus 1 and x plus 2. And then, we have here, okay, nakuha na natin yan. Now, ano naman daw ito? When we apply the common factor, what is the common factor of 12 and 4? 4. So, factor out natin yung 4. 12x divided by 4? 3x. Minus 4 divided by 4? 1. Now, can we do the cancellation? Yes. So, ano natin daw sa taas? 4. Sa baba? X. That's 2. Nasusundan po ba? Nasusundan? Okay, so dun muna tayo sa simple muna. Punta muna tayo sa common factor by common factor. Nakakaimigyan ba? Common of 5x minus 10. So 5x minus 10 divided by 5, 2. Here, divisible by 7. So we have 7x minus 2. 2. We'll cancel out. There's still 5 over 7. You got this answer. Okay, next. Here it is. What's the common of 4 and 12? 4. So we have 4. And then we have 4x divided by 4. x plus 12 divided by 4. 3. Here, their common is what? 8. So we have 8. Then we have x plus 3. We do the cancellation. So 4x. Can we do 4x? Yes. Divisible by 4. So that is 1 half. You got this answer. Okay, we will not stop unless we make sure that our fraction is already in simplest form. Okay, next. Okay, let's do this. So that will be x, x, so 15, so that is 5 and 3. So 5, 3, so we have here 5, positive times negative, negative 8, over x, x. Then we have 10, so that is 5 and 2. 5, 2, and then we have positive, negative, cancellation, na tira, x minus 3 over x minus 2. We got this answer. Okay, next, dito naman. So we have difference, so that is what? X plus 4 and X minus 4 over factoring trinomial. We have x, x, 4 is 4 and 3. Kasi 4 plus 3 is 7. So 4, 3, positive, positive, do the cancellation. So x minus 4 over x plus 3. Ayan na po yun. You got this answer. Okay. So that is about, okay, dividing functions. Okay, next we go to the composition of function. Can you do it? Yes. You can do it. Okay, composition of function. Okay, so we have here F, so we read it as, no fog, F, circle, G. Ano pong fog of fog? F, circle, G. Okay, so we have here, for the composition of function, So let's have here f circle g of x, it means f g of x. So it means that, let's say we have here an example, okay? So let's say we have here for our f of x, let's say our f of x is 2x plus 3. And then our G of X. is 5x minus 5. So when we say f circle g of x, it means that we're going to replace our x by the value of our g. Parang ipapasok natin. So it means that writing the x, we have here 2. instead of writing the variable x, we write the function g. So we have now 2 times 5x minus 5 plus 3. And then we perform. So 2 times 5x is 10x. Minus 10 plus 3. Nasusundan ko ba? Okay, again. From the given. Ito yung given natin ha. Yung function f. And function g. Tapos sabi dito, f, circle, g. So you go to the function f. Then instead of writing the variable x, we replace it by the function g. So now we have 2. Ito yun. So instead of x, isusulat natin yung function g. So 5x minus 5 and then plus 3. Taps up kung naisindihan. And then of course we do the process. 2 times 5x, 10x, minus 2 times 5, 10, and then plus 3. And then per 4, 10x minus 7. So this is f, circle, g. Now, let's have here g, see the difference, circle, f. This time, you go to function g. Okay. Papalitan naman natin yung x ng function f. So now we have 2x plus 3 minus 5. Nasusundan mo ba? Nasusundan. See the difference of x circle g and g circle f. Then we have 10x plus 15 minus 5. So that is 10x plus 10. Nasusundan po ba how it happened? Yes. Thumbs up po na kakasunod. Uulitin ko ha. Sa x circle g, you start in the function f. Eto. So replace our x, instead of writing variable x, we replace it by the function g. Dito naman po, you start in function g. Instead of writing x, You replace it by the function f. Nasusundan? perform the operation. Is it clear? Yes. Okay. Let's say we have here, so let's say another given pala. F of X is 2X plus 7. G of X is... Okay, 3x minus 9 and then we have h of x is 4x plus 3. Let us see. Kayo muna. Number 1, g circle h of x. And 3. Papalitan natin yung x na 4x plus 3. So 4x plus 3. 3. Then we have minus 9. Okay. Then we have 3 times 4x 12x plus 3 times 3 9. Minus 9. Pag magkamuka, magkaiba ng sign, cancelled. So the answer is 12x. Got this answer. Okay. Nakakasunod naman pala. Next. Dito naman tayo. Okay, puta ka sa function 8. So 4, papalitan natin yung x ng function f. So we have 2x plus 7 plus 3. Nasusundan ko ba? Then we have 8x plus 28 plus 3. So 8x plus 31. Got this answer. Okay, oh. You just need to look at it, it's here, right? Okay, next. There are still three of them, okay? Okay. That's how much, you're STEM, right? Okay, next. Let's see what we have here. Okay, f of x, let's put a little level up. 3x squared plus 5x. G of x. 4x minus 3. H of x. 2x squared minus 7. Okay, so let's have here G circle F. Palitan ang x na. 3x squared plus 5x. This is minus 3. This is 4x minus 3. Does that make sense? And then for 4. 4 times 3x, 12x squared. Plus 20x minus 3. That's it. You got that answer? Balit ka din natin ngayon. F circle G of X. F circle G. X ha? Okay, so we have 3. Ito daw, G. Ibig sabihin, eh naka-raise to 2 siya. So ibig sabihin, 4X minus 3 raise to 2. Iyan yung X raise to 2. Natusundan ba? Plus 5 Ito rin yung x 4x minus 3 Sino nakakuha ng ganitong pattern? Okay, so sold pa yan So now 3 Okay Paano daw tayo magmumultiply ng binomial? 4x times 4x Ano? 16x Square. It na naman. Minus 3 times 2. 6 times 4x. Oy. 24x. 3 times 2. 6 times 4x. 24x. Diba multiply them all? Remember? Doon sa circle ba yun? Okay. So we have minus 24x. Ito lagi plus. 3 times 3? 9. Ayan po yun. Plus, okay, 5 times 4x? 20. 20x minus 5 times 3, 15. Okay, ito naman. 3 times 16x squared. Ano daw? 48x squared. Minus 3 times 24x. 72x. Plus 3 times 9. 27. Okay, then, we put here plus 20x minus 15. So, 27 minus 15. Positive 12. Then, 72 minus 20. 52 negative because 72 is higher than 20. So, negative 52x. Let's just put this down. 48 x squared. And that's it. Who got that answer? Okay. We will replace the x with g. So we have times 4x minus 3. So raised to 2. Minus 7. Who has formed this? Who has? Only here. No one made a mistake. Can you understand? So let's multiply this. You can do the... 4x minus 3 times 4x minus 3. Pwede po yan. Nakakaintindihan po ba? Okay, now. 2. 4x times 4x. 16x. Ay, parang pares lang pala yun natin. Ay, okay. Oo nga pala. So we have here 16x squared minus 24x plus 9. Okay. Then minus 7. So, multiply natin 32x squared minus 48x plus 18 minus 7. So, we have now 32x squared minus 48x plus 11. Who got this answer? Okay. Okay, F, circle H, circle G. So, it means, you will do the F, circle H first. Okay, you will do the F, circle H first. Plus 5, so that will be what? 12 X plus 9 plus 5. So that will be 12x plus 14. Tapos, f, circle h, circle g. So ibig sabihin, mula ang year, papalitan niyo yung x ng g. Ganda lang po siya, kadali. 12 times, ano doon yung g natin? 2x minus 7 plus 14. So that will be 24x minus 84. Plus 14. Minus 84 plus 14. So that will be 24x minus 17. Ayan na po yung sagot. G circle F of x. So that is 2 times 3x plus 5 minus 7. So 6x plus 10. Minus 7, 6x plus 3. Then G, circle F, circle H. So, 6, pasok natin yung H. 4x plus 3 plus 3. Nasusundan ko ba? So, now we have 24x plus 18 plus 3. So, 24x plus 21. Who got this answer? Oh, you're so mean. You're sitting down. Now, let's give a raise to 2. F, 3x squared plus 2x. Okay, and then minus 4. So we have 6x squared plus 4x minus 4. G goes in. All of the x, we're going to change to g. So we have... 6 times x plus 5 raised to 2 plus 4x plus 5 minus 4. 6x times x, x squared. Plus 5 times 2, 10 times x, 10x. Plus 25. Plus 4x plus 20 minus 4. Then we have 6x squared plus 60x plus 150. Then we have 6 times 25, 150. Okay, and then, pasok natin itong 4x dito, tapos yung 20 dito. 20 minus 4, 16. Ayan na lang po. Then we have 6x squared. Plus 64x plus 166. Who got that answer? Oh, yung iba. Okay, tingnan nyo po ha? Aralin kung paano nangyari.

Okay, operations involving functions. Okay, so when we say operations, we are referring to the four fundamental operations that you have learned, right? And these are what?

Addition, subtraction, multiplication, and division. Okay, but may dadagdag po tayong isa and they add the last one is the composition of functions. So that will be added in the operations involving functions. So let's have here different functions. Let's say function f of x is 3x to the 4th plus 2x minus 7. Then we have function g of x which is negative 5x3 minus 9x plus 12. We have function h of x which is 12x to the fourth minus 3x raised to 3 plus 5x minus 10. And then we have the k of x which is 7x to the second power plus 5x minus 11. So as you can see we have four different functions.

So we're going to apply the operations but we start first in addition and subtraction. Then after that, we mix the addition and subtraction. So let's start from the basic. So let's do this one.

kindly add function f to function h number two subtract function k to function g okay so with your okay uh previous knowledge about how to add and subtract polynomials can you do it on your own without discussing it first let me see if you can still remember how to add and subtract polynomials. These are examples of polynomials, right? Okay, so simply add. Paano daw tayo nag-a-add? Balikan po natin yung ating learning from Okay, balikan po natin yung learning daw natin saan?

Sa rules of integers and sa ano pa po yun? Sa rules of integers at saka po sa exponent. How are we going to add if like sign? And if unlike sign, what happens to the exponent?

On the board, you want it on the board. Okay, number one. Okay, on that side. And any volunteer naman po for number two? Anyone po for number two who wants to show his or her answer po sa board?

Just give it a try. For number two po, who wants to find the sum and the difference naman po for number two? Number one po is sum because we are looking for the answer in addition. And number two naman is difference because we are looking for the answer in subtraction. Okay, let us check later.

Okay, if that will be the answer. How about for example two? For example two po, who wants to give a try? Subtract function k to function g. No one?

Okay, so let's us do first the number 1. Okay, so number 1. Okay, so we have here, okay, our function f which is 3x raised to 4 plus 2x minus 7 plus our function h which is 12x. to the fourth minus 3x3 plus 5x minus 10 okay so we have here okay so much better if we're going to that the terms in this uh in respective uh in respective group it is again combining like terms. So that is much better and more easier to see the sum of the function. Let's say we have here for our function f of x.

So we have here 3x to the 4th 3x raised to 4th Since we don't have a raise to 3, right? So, let's just leave an additional space for x raised to 3. Then we have plus 2x and then minus 7. Then we add the g of x, add the h of x. So, that will be what is our h.

So, 12x to the 4th, then minus 3x raised to the 3rd power plus 5x minus 3. As you can see, all the constants are in one line. Okay, all with x is in a similar line. Okay, and then we have x raised to 3 since we don't have x raised to 2. And then we have here our, those with x raised to 4 are also in a line. So now we have here, like sign, copy the sign, add the number.

Unlike, like sign, copy the sign. Add the number, copy the variable, and then bring it down kasi wala naman siyang partner. So negative 3x raised to 3, and then add 15x raised to 4. Okay, now, let's go to the example 2. This time, okay, function k subtract to function g. The same process din po, but... What we're going to do when it comes to subtraction, we chase the sign of the second function and then proceed to addition then.

Okay, example po, for number 2, we have function k, so we have 7x raised to 2 plus 5x minus 11. Subtract the function g. The function g automatically We will change the sign. Paltos subtraction. So from negative 5x raised to 3, it will be positive 5x raised to 3. Then, negative 9x will be positive 9x. And then, positive 12 will be negative 12. And then you add.

Okay, like sign, copy the sign, add the number. Copy the sign, add the number copy the variable bring down since walang sign to ibi sabihin nun ay positive hindi po po pwedeng walang sign na and then we have 5x raised to 3 so this is the difference po for number two nasusundan po ba thumbs up po if you can follow okay isa pa this time kayo nagagawa ah Number three. Okay, now let's... have H plus K.

And then number 4, in your place book, let's have, okay, F minus H. So for number 3, H plus K. And then for number 4, F minus H.

So do it in your place. X, Y, raised to 4 minus 3x raised to 3 plus 5x minus 10 and then add down natin yung function k so 7x raised to 2 plus 5x and then minus 11 as you can see okay naka-align po yung mga similar terms so like sign copy the sign Add the number plus 10, copy the variable, plus 7x squared, baba ng po, and the last one. So that will be the sum. You got this answer. Okay, next po, for number 4. Okay?

Pag ma-pinaba natin yung 7, you have to place po the plus sign. Kasi po, pag wala po yung plus sign, That will make your answer wrong. Kasi po, the number of terms po is depending on the plus and minus sign in between.

Do you understand? So we should put a plus sign because it will make your answer wrong. Number 4 So function f So we have here 3x to the 4th plus 2x minus 7 Okay, so space 4 raised to 3, raised to 2 Then, since it is minus and h preceded by the minus sign Change the sign of all function h.

We will change the sign automatic. So that will be negative 12x to the 4th, positive 3x raised to 3, negative 5x, and then plus 10. Then proceed to the addition. So now we have positive 3, because unlike sign, subtract, Copy the sign of the bigger number. Same with that.

Next one. And then since we don't have x raised to 2, we will just add x raised to 3. We don't have an answer that has space. So we have 3x raised to 3. Then we have, unlike sign, so subtract.

Copy the sign of 12. Negative 9x to the 4th. Who got this answer? Okay.

Hindi po tayo pwede mag-answer ng negative 9x to the 4th plus 3x3 minus 3x plus 3. Nagkakaintindihan ba? Kailangan po ay it must be consecutive in descending order. Sunod-sunod po. Wala pong bakante.

Nagkakaintindihan po ba? Okay, now. Who among you correctly answered number 3?

Raise your hand, be honest. Or yung iba, hindi pa rin nasundan yung number 3? Okay, put down your hands.

How about number 4? Who among you? Raise your hand. Okay.

So now let's go to number 5. Let us give a try kapag tatlohan na. Okay. So, number 5. Sa lang muna. Let's say we have here K minus H.

plus f. So k minus h plus f. Function k. So we have 7x raised to 2 plus 5x minus 11 minus function h.

So ibig sabihin, we change the sign of all function h. So we have here negative 12 X to the 4th plus 3x3 minus 5x plus 10. So, we can do this first. So, we have here negative 1. Cancel out. If it's the same, like terms with different signs, cancellation. And then bring down 7x squared.

Then we have positive 3x3 negative 12x to the 4th. Add do yung ating function f. So 3x to the 4th, positive 2x, and then minus 7. So we have now, this is positive 7x squared.

So we have now negative 8, positive 2x, positive 7x squared. Positive 3x raised to 3 and negative 9x to the 4th. Okay, who got this answer?

Okay. How about the others? Sanag kamali.

So, pwede niyo pong gawin by 2's muna. Tapos, yung sagot doon, ituktong natin yung F. Para hindi kayo nagrito, kung tatlo siyang mayroong line.

Kung hindi kayo sanay. Nakaka-intindihan ba? Okay, anyway, wala pong ano.

Okay, in what way po, kung anong gusto nyo, which is easier for you. Wala pong problema as long as you have the solution and give the correct answers. Okay, let's have another one.

Number six. Okay, so let's have here, this time. Okay, so we have...

H, H, minus F, minus G. of x. h minus f minus g of x.

Our function h, so that will be what? 12x raised to 4 minus 3 x raised to 3 plus 5x minus 10. Function f preceded by minus operation. So it will simply chase the sign of all function f.

So that will be negative 3x to the 4th, negative 2x and positive 7. Then add po muna natin. Unlike sign, subtract, copy the sign of the bigger number. Unlike sign, subtract, copy the sign of the bigger number. Okay, wala.

Then bring down, subtract 9x. to the fourth. Tapos, minus G. Yun G preceded by minus ole.

So change the sign of all G. So that will be negative, a positive, 5x3, positive, 9x, and minus 12. Like sign, copy the sign, add the number. Copy the sign, add the number.

Copy the variable. Since walang x raised to 2, it tatabi na. So positive 2x raised to 3, copy 9x to the 4th. So this will be the answer. You got this answer.

Okay. So it seems na, okay, more than half po nakakakuha. So, okay, yung iba po ay... Better ask help na lang po on how. Yung mga hindi nakakasunod.

Nakaka-incident ba? Okay, kasi hindi po pwede na. Paulit-ulit lang po tayo dito.

Meron pa po tayong tatlong operations na i-discuss. Okay, next. Now we go to multiplying function. So, bawatan lang po natin una para hindi masyadong burden sa inyo. Okay, ayan po.

Okay, ano naman daw yung ruling natin sa multiplying, diba? Okay, pag like sign, positive. Kapag unlike sign, negative. Multiply that number. Okay, and then add the exponent.

Nakakaintindihan po ba? Okay, let me see. Okay, paano daw tayo mag-multiply this time on your own?

It's like we're multiplying numbers. It's like 23 times 458. It's just like that. Okay, let's say we have here for number 1. Okay, function G times function F of X.

Then number 2, we have function K times function H of X. Let's see. Number 1, so we have G. of x, so we have negative 5x raised to 3 minus 9x multiplied by function f, which is 3x to the 4th plus 2x minus 7. So, what we're going to do to make it easier, it's easier to multiply by vertical. Okay, so what we're going to do, this first term, We will multiply all of these once at a time.

Can you understand? So we have sine first. Negative times positive is negative. Negative 5x3 times 3x to the 4th is what? Negative 15x7.

You add the exponent. We are not multiplying the exponent. Can you understand? Okay, it means negative 5 times 3, negative 15. X3 times X4, add the exponent.

Copy the variable, 3 plus 4 is 7. Isa-isa po. And then ito naman po, the next one. Negative 5X3 times 2X is what? Negative 10X to the 4th. Lagay natin dito, negative.

10x to the 4th Then ito uli Pangatlo Negative 5 x3 times negative 7, positive 35x raised to 3. So positive 35x raised to 3. Tapos ito naman, pagano naman din uli, isa-isa. Okay, negative 9x times 3x to the 4th. Negative 27x to the 5th. So negative...

27x to the 5th. Then, negative 9x times 2x. Negative 18x squared.

So, negative 18x squared. Negative 9x times negative 7. Positive 63x. Then you add.

So, positive 63x. Baba lang po natin. Baba lang po natin.

Baba. Then baba. Then baba. And that's it. Who got this answer?

Okay. Ilan lang po tumama? Dalawa. Okay.

Sige. Gabi niyo ulit yung number 2. So now let's go to number 2. Again, K times H. So what is our K? 7x squared plus 5x Multiply daw sa ating h which is 12x to the 4th minus 3x3 plus 5x Again, ito muna daw na 7x squared Isa-isa dito sa function h So we have here So what is 7 times 12?

84. So 84x, what will be the exponent of 3x? 6. 2 plus 4, 6. Then, next one. 7x squared times negative 3x3. 7 times negative 3. Negative 21. Then x squared times x3, x raised to 5. Then, ito naman.

7x squared times 5x is what? Positive 35x raised to 3. Okay, who got the first line? Okay, now we go to the second line. This one, again.

5x times 12x to the 4th. 60x to the 3rd. Okay, positive 60x to the 5th.

Ito, tsaka ito naman. So, 5x times negative 3x raised to 3. Negative 15x to the 4th. So, negative 15x to the 4th. 5x times 5x. So, positive 25x squared.

And then you add. Walang ka partner, baba. Baba din. 35, x3, baba. Negative 15, x to the 4th, unlike sign, so you subtract.

16 minus 21? 39. What will be the sign? Positive.

Positive. So positive, 39, x to the 5th, baba. 84, x raised to 6. Who got this answer? Okay, so next.

What if we have the one? Last one. Okay. Yung iba, saan pa rin nagkakamali?

Okay, sa exponent, anong ginagawa nyo sa exponent? Pinumultiply nyo? Oh, ina-add lang yan, diba?

Kapag wala po siya exponent, that means imaginary one. Okay, next po. Isan try lang muna, f times h of x.

f, so we have 3x to the 4th plus 2x minus 7 times do natin sa h, 12x to the 4th minus 3x3 plus 5x. So ito do muna, ito 3x raised to the 4th to all. Function of h.

So one at a time. So 3x to the 4th times 12x to the 4th. What is what?

36x raised to 8. x raised to 8. 3x to the 4th times negative 3x raised to 3. Negative 9x. 9x raised to 7. 3x to the 4th times 5x. Positive 15x to the 5th. So positive.

15x to the 5th. Ito naman kong pangalawa na. Dito ulit sa lahat na to. One at a time.

2x times 12x to the 4th. 24x to the 5th. Dito.

24x to the 5th. 2x times negative 3x to the 3rd. Negative 6x. to the fourth then 2x times positive 5x positive 10x squared.

We can add it first. So positive 10x squared minus 6x to the fourth positive 6 9x to the fifth minus 9x7 36x8. Let's go to the third one.

Negative 7 times 12x to the 4th. Negative 84x to the 4th. Negative 84x to the 4th.

Negative 7 times negative 3x raised to 3. Positive 21x raised to 3. So positive. So positive, what is this? 21. Where am I?

21x raised to 3. I just removed this. What is this? We just put it here. So positive 10x squared. Then we have, this again.

Negative 7 times positive 5x, negative 35x. Then add. So we have negative 35x positive 10x squared.

Saan? 30. Ay sorry po. 3, 9. So we have positive 21x3.

Then like sign, negative 70x to the 4th, a 90. X to the 4th, and then positive 3, 9, X to the 5th, let's do it again, negative 9, X, 7, and 36, X, 8. There you go. You got this answer. Okay, although it's long, but if you're going to do it one at a time, you won't get confused.

Do you understand? Minus F. What is our H? 16 X squared plus 11 X minus 19. Minus F. So change the sign of all turns of F. So we have negative 13 X to the fourth.

Negative 9 X 3. And then positive 15 X. Then you add first. So we have negative 19, positive 26x, positive 16x squared, negative 9x3, and negative 13x to the 4th. Add down natin yung k.

So we have positive 4x to the 4th, minus 13x squared, then we have 18. Then add. So negative 1, positive 26x, positive 13x squared, negative 9x3, and then negative 9x to the 4th. So this is the answer. Function G, negative 9x raised to 3 plus 11x. With minus 17x plus, so walang mapapalitang sign, 13x to the 4th plus 9. X3 minus 15x.

So add. So minus, ano do to? 32x.

Baba, positive 11x squared, cancelled. So 13x to the 4th. Tapos, minus H. Papagitan ng sign.

So negative 16x. X squared, negative 11x, positive 19, add. So positive 19, copy the sign.

So we have 43x. Then change the sign. Subtract the bigger number to the smaller number, copy the sign. So negative 5x squared.

X squared, tabi natin to, 13x to the 4th. Okay, negative 9x raised to 3 plus 11x squared. Multiply by, okay, f, 13x to the 4th plus 9x3 minus 15x.

Again, ito po muna. Negative 9 times 13, 13 times 9, 27, 117. So negative 117x to the 7th power. Then negative 9 times positive 9, so we have negative 81x to the 6th power.

Then negative 9 times 15. So that will be positive 135x to the 4th. Okay, next. Dito naman tayo sa pangalawa. 11 times 13. Diba pag phone na mumultiply sa dito, pag iniwalayin lang.

So that will be, okay, positive. One hundred forty-three. X to the sixth. Tapos eleven times nine.

Is positive nine hundred, ah, ninety-nine. X to the fifth. Positive ninety-nine.

X to the fifth. Eleven times nine. Two plus three, five. Then eleven times negative fifteen. Is negative 165x raised to 3. Okay, and then you add.

So negative 165x raised to 3. Positive 134x to the 4th. Positive 99x to the 5th. Okay, magkaiba ang sign. Subtract.

1, 4, 3, minus 81, 2, 14, minus 8 is 6. So that will be positive 62x6. Then we have, okay, 117x7. Ayan po.

We got it, 16x raised to 3. 2x times 13x, positive 26x raised to 2. Then we have negative 9 times negative 8, positive 72x raised to 2. Then negative 9 times 13, so we have 117. Okay, negative 117. Then add negative 117x. Add 19x squared below x3. I love you all.

Let's say we have here. Let's say I have here. Example po ah.

So we have x squared minus 9 over 3x plus. 9. Okay, now, can we simplify x squared minus 9 all over? over 3x plus 9. Can we do the cancellation here?

Pwede po ba tayo mag-cancel? Okay, ano? How are we going to cancel this one?

Pwede ba natin i-cancel yung 9 and 9 here? Oh. Okay, so we're going to get the factor first. So we cannot have, we cannot simplify this one. kasi po this is as one term kasi po we have the all over symbol or sign, nagkakaitin yan ba?

okay kung pag ganito po sana siya x squared over 3x minus 9 over 9, yan po we can have the cancellation pero since ito po ay all over, ibig sabihin hindi po natin siya pwede i-separate, lagi po siya counted as you 1. Okay, remember po yung ating, let's say we have here 9 over 12. Diba? This is not yet the simplest form. So we can still simplify it by getting the factor of 9, which is 3 and 3 and then 4 and 3. 3 times 3 is 9 and then 4 times 3 is 12. So pag kinasal natin, that makes our 3-4.

which is the simplest form of 9 over 12. Ang tawag natin doon is common factor. Okay, so gagawin natin, we're going to look for the common factor of the numerator and the denominator. So, isa-isa ni natin na, dito muna tayo sa binomial, hindi sabihin with two terms.

Dalawa lang po yung pwede natin gamitin dyan. Okay, here, as you can see, Paano daw tawag natin dito sa x squared, x to the fourth, x to the sixth, x to the eighth? Ano daw tawag natin dito? Perfect variable.

So pag sinabi natin perfect variable, it means that their exponents are even numbers. It can be divided into two. So what are the factors of x squared?

And that is x times x. How about x to the 4th? x squared times x squared.

How about x raised to 6? x 3 times x 3. How about x raised to 8? x raised to 4 times x raised to 4. Remember, kapag po like base, we simply copy the base and add the exponent. That's why we have copy the base x, 1 plus 1 is 2. Copy the base x, and then 2 plus 2 will be 4. This is what we call perfect variable.

Ang tawag po natin dyan ay perfect variable. Now, this one naman is what we call perfect number or perfect square. So what are the perfect number or perfect square?

So ito daw ay 1, 4, 9. 16, 25, 36, 49, and so on. What are the factors? Where did this 1 come from? 1 times 1. 2 times 2. 3 times 3. 4 times 4. 5 times 5. 6 times 6. What will be the perfect number after 36?

49. So that is 7 times 7. Okay, and then the operation is minus. What is the answer in subtraction? How do we call the answer in subtraction? Difference.

So, ang tawag po dito ay difference of perfect square. Okay. So, ang tawag po dito is different of perfect square. If the first term is perfect number or variable, and the second term is a perfect number or variable, and the operation is minus.

Okay. Another example we have here. Okay.

Is this a perfect difference of perfect square when we have this one? What do you think? Is this under difference of perfect square?

Yes, this is still under difference of perfect square. Why? Because 9 is...

perfect number x raised to the 4 is a perfect variable the operation is minus and then the 25 is a perfect number. If this becomes plus, it is not under difference. We don't have the sum of the perfect square.

Okay so if this is the given, it means what will be the factor of this one of number 1. That will be the square root of the first and second term and then two opposite signs. So, ganto po siya. X, X plus 3 minus 3. So, when we multiply this one, we will get X squared minus 9. So, here naman, paano daw yan?

That will be another ulit. Root of 9, 3, 3. Root of X raised to 4, X squared. square x square one positive one negative root of 25 five are you following?

okay let's have another one okay 16x6 minus 81y4 so what will be the factors of the following what is the factor? so that will be 16 4, 4, x3, minus 9, 9, and then y2, y2. When you multiply that, you will get 16x raised to 6, minus 81y raised to 4. Listen, so, if this is the given, Okay, this is not... A difference of perfect square.

So this time, pag ganito naman, ang possible po niyan ay common factor. So ano daw yung common factor? What will be the common factor of our 3x plus 9?

Okay, so the common factor is, okay, 3x plus 9, it means both terms is divisible by 3. So, factor out the 3. So, that is 3x divided by 3. The remaining is x. And 9 divided by 3 is 3. So, it means that this is 3 times x plus 3. And our sum is x plus 3 and x minus 3. Then, we do the cancellation. So now we can have x plus 3 over x plus 3, that is 1. So what's left in our numerator is x minus 3 and in our denominator will be 3. So that will be the simplest form. Nakakaintindihan po ba?

That is when given the difference of the perfect square and a common factor. Nakakaintindihan ba? That is applicable kapag dalawang binomial ang given. Gawa lang po ako ng given. Plus 5x minus 14 over x squared minus 49. Okay, this time, we have the trinomial and binomial.

Tandaan, kapag po binomial, there are two. possible lang po to find the factors. And that is if it is a difference of perfect square or there is a common factor. Okay, so dito sa baba, we all know na it is a difference of perfect square.

Now, we go dito po sa upper part. So, hindi po natin ipakia-apply yung difference of perfect square kasi that is only applicable in binomial. Since it is trinomial, there is no... po silang common factor. Di po ba?

So out yung dalawang yun. So as you can see, we can have factoring trinomial. So paano naman daw natin ipa-factor ang trinomial? So we have here x squared plus 5x minus 14. So yung ating pong last term, pupunin natin yung possible factors.

Ibig sabihin, Numbers that when we multiply will produce 14. And that is 7 times 2 and 14 times 2. Now when you add or subtract, it will produce the middle term. So the possible is 7 and 2 because 7 minus 2 is 5. So 14 cannot produce the 5. So it means that the possible of Doña is 7 and 2. Then we have the root of X. word, so that will be x and x. Since it is minus, it means it's an opposite sign. Can you understand?

Okay, then, since our middle term is positive, it means the one that is positive is the bigger number. 7 is bigger than 2, it means this is positive, this is negative. Always the bigger number, it always looks like Bye. yung sign ng middle term.

Nakakaintindihan po ba? So, ito sabihin, when you multiply x plus 7 to x minus 2, the product is x squared plus 5x minus 14. So, ito daw yung factors niya. So, we have here x plus 7 and then x minus 2. Then, alam na natin yung factors ng baba.

Ano doy factors ng baba? Okay, x you Plus 7 and x minus 7 because it is difference of perfect square. So we have now x plus 7 and x minus 7. Can we do the cancellation?

Yes. Yes, here it is. So it's left at the top, x minus 2. It's left at the bottom, x minus 7. There's the answer. Can you understand? Okay, next.

May isa pa po. Kumbaga, ano na yung nakuha natin? Okay, common factor, difference of perfect square, and factoring trinomials. Okay, next. Let's have another one.

So, let's have 12x. Minus 4 over 3x squared plus 5x minus 2. Okay, so we have here 12x minus 4 over or divided by 3x squared plus 5x minus 2. So, what is the possible here? Common factor or difference of perfect square.

Is this difference of perfect square? No. So, ang possible lang is common factor.

Okay, now, can we apply what we did kamina dito? Can we apply yung makukuha natin daw yung middle term? By the factors of the last term? No.

Because there is a coefficient beside x squared. Ang tawag naman po natin dito, If this is not there, we can use it as a factoring trinomial. If possible. Can you understand? But if there is a numerical coefficient here, what we call is the process of guess, check, and revise.

It means, we will think of a number, we will fix it, we will check if it will be correct, if not, we will rearrange it again. We will check again if it will be correct. Tapos, rearrange ulit natin hanggang sa tumama. Ayun po yung guess, check, and revise.

Okay. Huwag na po tayo pumunta ng quadratic formula. Ang tagal nun.

Nagkakaintindihan ba? Okay. Now, let's go to this one. 3x squared plus 5x minus 2. So, dalawa ulit daw yung kanyang factors. Okay, 3 means 3 times 1. Wala nang iba, diba?

So, we can have 3 here and 1 third. Tapos itong x, tag-isa. Hindi ko mo pwede na yung isang first term wala pong variable. Kailangan po, pareha silang may variable.

Okay, so now, we have 2. So, dito maglalaro tayo sa 2. Ang 2 natin, it can be 2 times 1 lang. Wala nang iba. Okay, so let us place 2 here. and wonder then as you can see negative doto so it is a bn makaiba sila ng sign okay now let's Let's say we make it positive and we make it negative.

Kasi sabi magka-opposite side. Hindi na po tayo mag-iisip na magpo-produce itong 5. Dapat mag-produce yung 5 kapag tinimes po natin itong ating inner sa outer. Tapos inad natin, lalabas dapat yung middle term. Okay. 2 times x.

2x. Positive 2x. 3x times negative 1, negative 3x. If we add it, that makes negative x.

If we add this, 2x minus 3x, negative x. Did we get the middle? No. We changed the sign.

Negative times positive, so it will be negative 2x, positive, positive x. Hindi pa rin. We change the number. 1 and 2. Paritan natin. So we have negative 1 times x.

Negative x. 3x times 2. 6x. Negative x plus 6x when we add. Positive 5x.

Check. Nagkakaintigyan ba? So hindi sabihin.

Ang factor ng 3x squared plus 5x minus 2 is 3x minus 1 and x plus 2. That is what we call guess, check, and revise. Okay, so now, nakuha na natin ito. Ang factor doon ito ay 3x minus 1 and x plus 2. And then, we have here, okay, nakuha na natin yan. Now, ano naman daw ito? When we apply the common factor, what is the common factor of 12 and 4?

4. So, factor out natin yung 4. 12x divided by 4? 3x. Minus 4 divided by 4? 1. Now, can we do the cancellation?

Yes. So, ano natin daw sa taas? 4. Sa baba? X. That's 2. Nasusundan po ba?

Nasusundan? Okay, so dun muna tayo sa simple muna. Punta muna tayo sa common factor by common factor.

Nakakaimigyan ba? Common of 5x minus 10. So 5x minus 10 divided by 5, 2. Here, divisible by 7. So we have 7x minus 2. 2. We'll cancel out. There's still 5 over 7. You got this answer. Okay, next.

Here it is. What's the common of 4 and 12? 4. So we have 4. And then we have 4x divided by 4. x plus 12 divided by 4. 3. Here, their common is what?

8. So we have 8. Then we have x plus 3. We do the cancellation. So 4x. Can we do 4x? Yes.

Divisible by 4. So that is 1 half. You got this answer. Okay, we will not stop unless we make sure that our fraction is already in simplest form.

Okay, next. Okay, let's do this. So that will be x, x, so 15, so that is 5 and 3. So 5, 3, so we have here 5, positive times negative, negative 8, over x, x. Then we have 10, so that is 5 and 2. 5, 2, and then we have positive, negative, cancellation, na tira, x minus 3 over x minus 2. We got this answer. Okay, next, dito naman.

So we have difference, so that is what? X plus 4 and X minus 4 over factoring trinomial. We have x, x, 4 is 4 and 3. Kasi 4 plus 3 is 7. So 4, 3, positive, positive, do the cancellation. So x minus 4 over x plus 3. Ayan na po yun. You got this answer.

Okay. So that is about, okay, dividing functions. Okay, next we go to the composition of function. Can you do it? Yes.

You can do it. Okay, composition of function. Okay, so we have here F, so we read it as, no fog, F, circle, G. Ano pong fog of fog? F, circle, G.

Okay, so we have here, for the composition of function, So let's have here f circle g of x, it means f g of x. So it means that, let's say we have here an example, okay? So let's say we have here for our f of x, let's say our f of x is 2x plus 3. And then our G of X.

is 5x minus 5. So when we say f circle g of x, it means that we're going to replace our x by the value of our g. Parang ipapasok natin. So it means that writing the x, we have here 2. instead of writing the variable x, we write the function g. So we have now 2 times 5x minus 5 plus 3. And then we perform.

So 2 times 5x is 10x. Minus 10 plus 3. Nasusundan ko ba? Okay, again.

From the given. Ito yung given natin ha. Yung function f. And function g.

Tapos sabi dito, f, circle, g. So you go to the function f. Then instead of writing the variable x, we replace it by the function g.

So now we have 2. Ito yun. So instead of x, isusulat natin yung function g. So 5x minus 5 and then plus 3. Taps up kung naisindihan.

And then of course we do the process. 2 times 5x, 10x, minus 2 times 5, 10, and then plus 3. And then per 4, 10x minus 7. So this is f, circle, g. Now, let's have here g, see the difference, circle, f.

This time, you go to function g. Okay. Papalitan naman natin yung x ng function f. So now we have 2x plus 3 minus 5. Nasusundan mo ba? Nasusundan.

See the difference of x circle g and g circle f. Then we have 10x plus 15 minus 5. So that is 10x plus 10. Nasusundan po ba how it happened? Yes.

Thumbs up po na kakasunod. Uulitin ko ha. Sa x circle g, you start in the function f. Eto. So replace our x, instead of writing variable x, we replace it by the function g.

Dito naman po, you start in function g. Instead of writing x, You replace it by the function f. Nasusundan?

perform the operation. Is it clear? Yes. Okay. Let's say we have here, so let's say another given pala.

F of X is 2X plus 7. G of X is... Okay, 3x minus 9 and then we have h of x is 4x plus 3. Let us see. Kayo muna. Number 1, g circle h of x. And 3. Papalitan natin yung x na 4x plus 3. So 4x plus 3. 3. Then we have minus 9. Okay.

Then we have 3 times 4x 12x plus 3 times 3 9. Minus 9. Pag magkamuka, magkaiba ng sign, cancelled. So the answer is 12x. Got this answer.

Okay. Nakakasunod naman pala. Next.

Dito naman tayo. Okay, puta ka sa function 8. So 4, papalitan natin yung x ng function f. So we have 2x plus 7 plus 3. Nasusundan ko ba? Then we have 8x plus 28 plus 3. So 8x plus 31. Got this answer. Okay, oh.

You just need to look at it, it's here, right? Okay, next. There are still three of them, okay? Okay.

That's how much, you're STEM, right? Okay, next. Let's see what we have here.

Okay, f of x, let's put a little level up. 3x squared plus 5x. G of x. 4x minus 3. H of x.

2x squared minus 7. Okay, so let's have here G circle F. Palitan ang x na. 3x squared plus 5x. This is minus 3. This is 4x minus 3. Does that make sense? And then for 4. 4 times 3x, 12x squared.

Plus 20x minus 3. That's it. You got that answer? Balit ka din natin ngayon. F circle G of X. F circle G.

X ha? Okay, so we have 3. Ito daw, G. Ibig sabihin, eh naka-raise to 2 siya. So ibig sabihin, 4X minus 3 raise to 2. Iyan yung X raise to 2. Natusundan ba? Plus 5 Ito rin yung x 4x minus 3 Sino nakakuha ng ganitong pattern?

Okay, so sold pa yan So now 3 Okay Paano daw tayo magmumultiply ng binomial? 4x times 4x Ano? 16x Square.

It na naman. Minus 3 times 2. 6 times 4x. Oy.

24x. 3 times 2. 6 times 4x. 24x.

Diba multiply them all? Remember? Doon sa circle ba yun?

Okay. So we have minus 24x. Ito lagi plus.

3 times 3? 9. Ayan po yun. Plus, okay, 5 times 4x? 20. 20x minus 5 times 3, 15. Okay, ito naman.

3 times 16x squared. Ano daw? 48x squared. Minus 3 times 24x. 72x.

Plus 3 times 9. 27. Okay, then, we put here plus 20x minus 15. So, 27 minus 15. Positive 12. Then, 72 minus 20. 52 negative because 72 is higher than 20. So, negative 52x. Let's just put this down. 48 x squared.

And that's it. Who got that answer? Okay. We will replace the x with g. So we have times 4x minus 3. So raised to 2. Minus 7. Who has formed this?

Who has? Only here. No one made a mistake.

Can you understand? So let's multiply this. You can do the...

4x minus 3 times 4x minus 3. Pwede po yan. Nakakaintindihan po ba? Okay, now. 2. 4x times 4x. 16x.

Ay, parang pares lang pala yun natin. Ay, okay. Oo nga pala.

So we have here 16x squared minus 24x plus 9. Okay. Then minus 7. So, multiply natin 32x squared minus 48x plus 18 minus 7. So, we have now 32x squared minus 48x plus 11. Who got this answer? Okay. Okay, F, circle H, circle G.

So, it means, you will do the F, circle H first. Okay, you will do the F, circle H first. Plus 5, so that will be what? 12 X plus 9 plus 5. So that will be 12x plus 14. Tapos, f, circle h, circle g.

So ibig sabihin, mula ang year, papalitan niyo yung x ng g. Ganda lang po siya, kadali. 12 times, ano doon yung g natin? 2x minus 7 plus 14. So that will be 24x minus 84. Plus 14. Minus 84 plus 14. So that will be 24x minus 17. Ayan na po yung sagot. G circle F of x.

So that is 2 times 3x plus 5 minus 7. So 6x plus 10. Minus 7, 6x plus 3. Then G, circle F, circle H. So, 6, pasok natin yung H. 4x plus 3 plus 3. Nasusundan ko ba?

So, now we have 24x plus 18 plus 3. So, 24x plus 21. Who got this answer? Oh, you're so mean. You're sitting down.

Now, let's give a raise to 2. F, 3x squared plus 2x. Okay, and then minus 4. So we have 6x squared plus 4x minus 4. G goes in. All of the x, we're going to change to g.

So we have... 6 times x plus 5 raised to 2 plus 4x plus 5 minus 4. 6x times x, x squared. Plus 5 times 2, 10 times x, 10x. Plus 25. Plus 4x plus 20 minus 4. Then we have 6x squared plus 60x plus 150. Then we have 6 times 25, 150. Okay, and then, pasok natin itong 4x dito, tapos yung 20 dito. 20 minus 4, 16. Ayan na lang po.

Then we have 6x squared. Plus 64x plus 166. Who got that answer? Oh, yung iba.

Okay, tingnan nyo po ha? Aralin kung paano nangyari.

Transcript for:Functions Operations and Factoring

Transcript for:
Functions Operations and Factoring