In this video, we will discuss how are we going to find the greatest common factor or what we call GCF. Of course, by listing or prime factorization. Second, we will factor polynomials completely.
with common monomial factor and use distributive property to factor polynomials. So before we proceed to the discussion, let us unlock first greatest common factor or yung tinatawag nating GCF. Ano nga ba ang GCF? Ang GCF, ito yung mga factors na common dun sa mga terms.
Take note. Kapag kumukuha tayo ng common factors nila, kukunin natin yung pinakamataas. Kaya from the word itself, greatest.
So kukunin natin yung pinakamataas na factor nila. Pangalawa, prime factors. Pag sinabi nating prime factors, these are numbers, ito yung mga numbers na ang tanging factor niya lang ay 1 and the number itself. Halimbawa, 2. So ang factors lang niya ay 1 and itself. 3 and 5. Those are some of the examples.
Next, monomial. Pag sinabi nating monomial, this is a special kind of polynomial na kung saan ang bilang ng terms niya ay isa lang. So, from the prefix, mono means one. So, we only have one term for it.
And then, polynomial. This is an expression combined with... constant.
Ito yung pinagsama-samang constant, variable, or pwede rin product of two. Product ng constant or variable. Or product ng constant and variable.
But take note, dapat ang exponent niya must be a whole number. So, hindi tayo pwedeng magkaroon ng negative exponent doon. Fraction or yung may symbols. Next, standard form.
When we say standard form, this is a kind of arrangement of a polynomial in descending order. When we say descending order, it starts at the highest degree to least degree, the lowest. Okay, greatest common factor. It refers to the common factor having the greatest numerical factor. and with variables having the least degree.
So, ang kukunin natin na numerical factor ay yung pinakamataas at sa variable naman tatandaan na ang kukunin natin ay yung pinakamababa ang degree or exponent. Example, meron akong 6x squared and 15x raised to 4. So, paano natin kukunin ang GCF nito? So by listing, we can do this.
So the factors of 6 are 1, 2, 3, 6, and x squared. In 15, we have 1, 3, 5, 15, and x raised to 4. So how do we get GCF? What is their common factor? That is 3. Take note, it should be the greatest. And then, sabi nga, sa pagkuhan ng variable, kukunin natin yung pinakamababaang degree.
Since meron tayo ditong dalawang x, so kukunin natin yung pinakamababaang degree. And that is x squared. So therefore, we will have 3x squared.
Next, another way is by prime factorization. So 6x squared. So mag-iisip ka ng mga factorization. factors of that number or term that is just a prime number.
So for example, 6. So you cannot use 6, 1 and 6 because 6 is not a prime number. So you will think of numbers that are just prime numbers. So for example, in 6, we have 2 and 3. 2 times 3 is 6. 2 and 3 are prime numbers.
And then x squared. So for 15, hindi na tayo pwedeng gumamit ng 1 and 15. We can only use 3 and 5 if you are using prime factorization. And then, x raised to 4. So, anong common sa kanilang dalawa? We have 3. and then we will get the lowest exponent in the variable and that is squared.
So we will have the GCF is 3x squared. Next, let's try to find the GCF of each pair of monomials. So I have here 6a and 18ab. So for 6a, by using prime factorization, we have 2 times 3 times a. For 18ab, we have 2 times 3 times 3 times a times b.
So, uulitin ko nga, hahanap or mag-iisip lang tayo ng mga factors na prime numbers lang po. So, 2 times 3 is 6, 6 times 3 is 18. Now, anong meron sa 6a na meron kay 18ab? Pareho silang may 2 and 3. Pareho rin silang may A. So, yun ang ibababa natin. We will have 2 times 3 times A.
And then we will multiply 2 times 3. That will become 6A. Next, I have here 10A and 12A squared B. So, hanap ulit tayo o mag-isip tayo ng mga prime numbers na factors ng 10. So, we have 2 and 5. 2 times 5 is 10, and then A.
So for 12A squared B, we have 2 times 2 times 3. Because 2 times 2 is 4, 4 times 3 is 12. And then A squared times B. So what's common between the two of them? We have 2. And then, they both have A. But remember, if they both have a variable, then...
That is their common variable. Pero kapag ganun, kukunin lang natin yung may pinakamababa ang exponent. So we will get a, not a squared. So bababa na natin yun.
We have 2 and then a. Now, bakit hindi ko nilagay si 3? Because wala namang 3 sa 10a. Bakit hindi ko nilagay si 5? Kasi wala namang 5 si 12a squared b.
Bakit hindi ko nilagay si b? Kasi wala namang b kay 10a. So we will only get 2 and a.
And that is, the GCF is 2a. Next, I have here negative 8x squared y and 16xy. So for negative 8, since it's negative, we have a negative sign. And then 2 times 2, 4 times 2, that is 8. X squared times Y.
For 16XY, we have 2 times 2, 4. Times 2, 8. Times 2, 16. And then X and Y. So what's common between the two? We have 2 times 2 times 2. We won't add 2 to 16XY. kasi tatlo lang naman yung 2 dun sa negative 8x squared y. Now, pareho silang may common na x variable.
So again, kukunin lang natin yung pinakamababa ang exponent, and that is x, same as y. So bring down natin yung tatlong 2, ibaba natin si x at si y. So we will have now 2 times 2, 4 times 2, that is 8xy.
Next, 8ab raised to 3 or 8ab cubed and 10a squared b squared. So for 8ab cubed, we have 2 times 2 times 2 times a times b cubed. For 10a squared b squared, we have 2 times 5 times a squared times b squared. So anong common? We have 2. We will not get the other 2s kasi wala naman ng 2 dun sa 10a squared b squared.
Hindi rin natin pwedeng ilagay si 5 kasi wala namang 5 kay 8ab cubed. Now, for a variable, kukuni natin yung mababang exponent and that is a. For b naman, ang mababa ay si b squared.
So, yun ang kukuni natin. So, we will have now 2a and b squared. So the GCF is 2ab squared. Okay, let's proceed on how are we going to factor.
polynomial. Factoring polynomial refers to rewriting a polynomial as a product of polynomials of smaller degree. So we will try to factor the given polynomial using the GCF. So I have here 4x squared plus 6x.
So we will take the GCF that is 2x In factored form, we will put the GCF outside. Now, how do we get what is inside the parenthesis? Okay, so we can think of a number or think of a monomial. If you multiply, the resulting product will be 4x squared.
So, like this, 2 times 2 is 4. And then, 2x, we will add the exponent. That's why it became... x squared. Now, bakit 3? Because 2 times 3, ayun ang 6. And then since isa lang ang x nya, that's why we have 6x.
Another way, pwede rin namang si 4x squared i-divide mo sa 2x. 4 divide 2, that is 2. x squared. squared divided x, kapag nagdi-divide tayo, we are subtracting the exponent if they have the same base. That's why x squared minus 1, ito yung 2x dun sa GCF natin, kaya naging x na lang dun. And then, 6 divide 2, kaya nakuha natin si 3. Another example, 3x squared plus 6x, the GCF is 3x.
So lalagay natin si 3x sa labas. Paano natin kukunin yung nasa loob? Think of a monomial again.
So 3x and then since x squared yun, so maglalagay tayo ng isang x. And then 3 times 2, that is 6. And then we only have 1x kaya 6x dun. Next! 6x raised to 4 minus 14x squared. So the GCF is 2x squared.
We will put it there. How will we get 3x squared? Because 2 times 3 is 6. x squared and then x squared, that will become x raised to 4. Now, this is minus.
And then 2 times 7, that makes it 14. And then since we only have x squared, that's why we have 14x squared. Okay, let's have the detailed solution. Use GCF to rewrite the polynomial. So, kagaya lang din to kanina.
Pero ipapakita natin this is another way using distributive property. So, eto, ilagay si GCF si 2x. And then use distributive property, ganun ulit, ano yung itatimes mo kay 2x for you to get 4x squared, at ano yung itatimes mo rin dun sa second term mo para makuha mo si 6x. So, ayun, paano natin to gagawin?
Kung ano yung GCF, lagay natin dun sa labas, and then kung ano yung nakuha mong common monomial factor. rather, yun ang ilalagay natin sa parentheses. Next, 3x squared plus 6x.
So again, use the GCF to rewrite the polynomial. Ilabas natin si 3x and then 3x dun sa second term. Think of a monomial factor which is x and 2 sa kabila. So we will have now 3x times x plus 2. Paano naging ganyan?
Write the GCF and then the common monomial factor. Next, 6x raised to 4 minus 14x squared. So, same procedure.
Baba si GCF and then enclose by the parenthesis or factor natin yung 3x squared. Copy the sign and then bring down 7. Next, we have here 7a times a plus 3 minus c times a plus 3. So, paano natin siya ipa-factor? So as simple as, tingnan sa kung ano yung GCF nila. Pag sinabing GCF, ano yung meron sila pareho?
That is A plus 3. So ibababa natin yun. And then, yung 7A minus C, ibababa rin natin. Ayan. Okay, another example. So kunin ulit yung GCF, yung common sa kanila, that is N plus 3. And then, 8m plus b, that is another binomial.
So, that's where we can factor. Okay, another example. I have here 4 times 3b minus 1 plus 5b times 1 minus 3b plus 4c times 3b minus 1. So, if you notice, Paano natin itong mapafaktor? Eh yung nasa loob ng parenthesis dun sa middle term natin ay 1 minus 3b. So wala silang GCF.
Diba? So mahirap. So paano ang gagawin natin? We all know that 1 minus 3b is equal to negative times the quantity of negative 1. plus 3b. So, paano natin gagawin yun?
Ita times natin siya by negative 1. Kaya ang mangyayari magiging negative 5a times negative 1 plus 3b plus 4c times the quantity of 3b minus 1. So, ayan na yung nangyari. So, simula dun sa taas. We just times it by negative 1, so it becomes negative 5a. 1 becomes negative 1. Negative 3b becomes positive 3b. Now, they are still not the same.
So, what are we going to do? We will just rearrange the given polynomial. So, we can now fix it.
So, what are we going to do? We are just going to fix it. If you put it on the left, you put the minus 1 on the right. So, if you notice, they are the same as GCF. So, we will now have 3B minus 1 times 4 minus 5A plus 4C.
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