[Music] Hello everyone. Welcome back to our Matth is Ip YouTube channel. Our topic for today is all about multiplication and division of binomials and multinomials. But before that, if you're new here, don't forget to subscribe for more lessons. And for our teachers, if you need our editable PowerPoint presentations, you can check the link in the description below or message us on our Facebook page. Okay, so without further ado, let's begin with our lesson. Okay, so this is a continuation lesson on our previous topic on multiplication and division of monomials. If you haven't watched that, if you haven't watched our previous video lesson about this topic, watch it first so you can better understand our topic for today. You can click the link or you can just check our previous video on our playlist. Okay. Our competence is to multiply simple monomials and binomials with simple binomials and multinomials using the distributive property with various techniques and models like for example the foil method and the vertical method later on. Okay. So our lesson for today is divided into three parts. This is in three parts. This is the first one. Multiplying binomials and other polynomials by a monomial. So after this and then the second part we will have an activity or an exercise for you to practice. So we can proceed with the third part. Okay? So let's begin. I will give you three examples on this and then later on you on your own. So first we have here multiplication of monomial by binomial. Okay. So let's just review. Remember monomial is a polynomial with one term. Just like here in this example we have 3x. This is the monomial. And then binomial is a polynomial with how many terms? Yes, two terms. That's why we have here an example x + 4. So again, monomial 3x multiplied to a binomial which is x + 4. Two terms separated by plus or minus. Okay? So what do we do here? We are just going to distribute the monomial to the polynomial. Okay. So, this is our monomial. We'll just distribute them one by one here in terms of the binomial. So, 3x x first or the first term of our binomial. So, we just multiply them. So, 3x * x. And then after that, we'll also multiply 3x by the second term of our binomial. So here we will have 3x * 4 and then we just carry out the addition operation. Okay. Then the next step is to apply the distributive property. Meaning let's just multiply. Okay? Let's just operate on them. So we have here 3x x will be 3x s. Okay. Why did his exponent become two again? Remember in our previous video our discussion on multiplying monomials. When we multiply two variables with the same base, all we have to do is add their exponents. So what is the exponent of x here? Yes, there is one exponent here and there's another one exponent here. Let's just add them so it becomes 2. Therefore, x^ 2. After that, we have here 3x * 4. The answer is 12x. And the final step is to combine similar terms if necessary. Only if necessary. So do we have similar terms here? We have 3x² and then 12x. Are they similar? Okay. No, please. Although they have the same variable, the exponents are different. So they are dissimilar. Therefore, this will be our answer. Okay. Example number two, we have here -3x the quantity of 4x - 2y + 8. So this is now an example of monomial times a trinomial. Okay. Remember trinomial is composed of three terms by plus or minus. Okay. So how do we do this? Again this is also distributive. Okay, that means we'll distribute them one by one. So we have here -3x. Let's first multiply it by 4x. The first term of the trinomial. So -3x * 4x. This is it. Next one we will multiply -3x again by the second term this time which is -2y. Okay? So it's going to be -3x -2y, we'll just add them. Okay, let's just add what their product will be. Next one, -3x +8. Therefore, we have here -3x * 8. Okay, let's just add the products. Okay, next is we can now simplify -3x * 4x we will have -1x s. After that we have here -3x - 2y it will be 6xy. Okay? Remember negative negative is positive. Okay? Take note, we did not add exponents for x and y because their bases are different. So all we have to do is to just copy x and y without adding their exponents. Okay? This is not the same as x here, it's the same as x, so we added the exponent. And then last we have -3x 8 we will have -24x. Remember negative positive is negative. Okay? Why did our operations disappear ? Okay? What you can do is determine what the final answer will be for each product, what its final sign will be , that will also be the operation in the middle. Like for example here a negative times negative is positive. That will be his operation. Here, negative times positive is negative. That's also what he'll be doing here, which is a minus. Okay, so it's easy. Next, can we still simplify this further? Are there any other similar terms we can add here? Okay, that's it . Therefore, this is now our final answer. Okay. In example number three, we have here monomial times multinomial. So if we have here a s b c multiplied by this polynomial. If you notice, he's not just two terms. We don't just have three terms, we have four terms here. So this is what we call it if more than three we can just call this multinomial or we can also say polynomial in general. So how do we multiply this? So just like the same in our previous examples, all we're going to do is just distribute our monomial. So the rules are the same. So let's begin. Let's now distribute a s b³ c times our first term 2ab. Okay? This is it. Next, we will now distribute a s b³ c times our second term here in this multinomial which is -4c. Yes, that includes the negative side as well. This will be his sign, the operation. Okay. Next, let's multiply the monomial by our third term which is 8C. Okay, this is it. And then last our monomial to our fourth term which is -2. Okay, there you go. So again remember, we will only use addition to add or combine all our products. Let's now multiply this one by one. So first we have here a s b cu c 2ab. Do we have similar terms between these two? Okay, there is. So, we can add their exponents. However, for our numerical coefficient, we only have two here. That's why our answer is 2. Okay, so 2 a³. Why? We added the exponent of a here to the first monomial and then the second monomial. So 2 + 1. Next one we have here b^ 4 because 3 added to 1 here. And then lastly, let's just copy and paste the C. There you go. Okay. Next let's multiply a s b³ c to -4BC. So take note this is positive times negative. Therefore our answer is negative. That's our operation too. He will be minus. So - 4a s b^ 4c s by adding all their exponents. Okay. Next one we have here is our monomial times the third term which is 8c. Okay. Since there is no numerical coefficient here, we just copy the 8 and then we have we will copy a squ b cube and then the c, that's the only thing similar between the two of them. We just add their exponents. Okay. And then lastly, there are no similar terms between these two. Therefore, let's just combine them. So, we have here -2 a s b³ c. Okay. Lastly, let's see if there are any similar terms here. This means that the variables are the same and at the same time the exponents are the same. Is there any ? Okay. Nothing. If nothing else, that will be our final answer. Okay. Now let's go to the second part of our lesson which is division. So dividing binomials and multinomials by a monomial. So let's begin with some examples. Okay. First we will have an example of dividing a binomial by a monomial. Okay. So we have here binomial 8x² + 4x cu / 4x. Okay. So how about this? Okay. All we have to do here is to divide each term of the dividend to the given divisor. So we will divide each term by our divisor, which is the monomial. Okay? So since there are two terms here, we will have two divisions. Okay? So let's start with the first part. 8x is our first uh term of our binomial, we will divide it by 4x. And then after that we will add it by dividing 4x cux anyway. Okay? So we just divided him. So our answer here is 8x² / 4x, which is 2x. Remember in our previous lesson, when we divide monomials, all we have to do is to divide the numerical coefficients. So we have here 8/ by 4. The answer is 2. How about the variables? If our variables are the same, they are similar, similar base, all we will do with our exponents is subtract them. Okay? So that's the opposite of multiplication. Where we added earlier, now we're going to subtract. Okay. So we have here 2 minus the exponent of x here in the denominator which is 1. So 2 - 1 is 1. Okay? That's why he became 2x. Next one. Here, 4 / 4 is 1. And then we have here x^ 3 / x. So let's just subtract the exponent. 3 minus the exponent here at the bottom which is 1. We will have x s. Okay. So if we can't simplify it anymore, we can't combine it anymore because they are dissimilar terms, this is our final answer. Okay. For example number two, we have here a trinomial divided by a monomial. So it's similar to earlier, our polynomial above, we'll just divide it into individual monomials so that we can divide it more easily. So we will separate these three terms into three separate uh monomials. Okay? That's so we can divide by our divisor. Okay? So take note of our operation. Okay? He's still the same . So it's still a minus here and then it's still a plus here. At the same time, whatever the operation here is, it seems like that's also the sign of our numerator. So if this is subtraction, it means that our numerator here, 9A cu is a negative. Okay? That seems to be his sign. It's the same here. If this is plus, the sign of 6a s is positive. Okay. So this will be useful when we divide. Alr. So when we divide, our answer will be - 4a³ + 3a - 2. Okay. Let's explain. First we have here 12 / -3. That's why it's -4. And then we have here a^ 5 / a^ 2. Okay. Next one we have here -9. Okay, -9 because our operation is included in that. -9 / -3 -/3 is +3. Okay? And then we have 3 - 2. So our exponent here is 1. We have here + 6 / -3. / nega is negative. Okay. So, + 6 / -3 is 2. And then since our numerator is the same, which is a sator and a sen, we can just cancel this out. Okay. Therefore, our final answer is -4a cu + 3a - 2. Okay. For our last example, we have here another trinomial divided by a monomial. Okay. So again, we'll divide him into three separate parts so we can divide him more easily. Okay? So we will divide by the same divisor or the same monomial. Okay? And as we have said a while ago, we have here the operations. That will also be our operations here. Minus and then plus. And then for our final answer, we have here -4x² y³ - xy + 3y. Okay, let's explain one by one. So we have here -36 / 9. The answer is -4. Okay. And then x^ 4 / x^ 2 4 - 2 is 2. And then we have here y 4 / y^ 1 4 - 1 is 3. That's why another one we have here 9/ 9 this will be 1. So we don't need to write the numerical coefficient. Next 3 - 2 is 1 and then 2 - 1 is 1. That's why we have here - xy. Another one we have here 27 / 9 is 3. And then this is x² / x². You can cancel this out because it's the same . And then we have here y S / Y 2 - 1 is 1. Okay? So this is our final answer. Okay. Before we move on to the third part of our lesson, we have an activity here so you can practice multiplication and division of polynomials involving monomials. Okay. So we have here multiplication for set A and then division for set B. Okay? You can pause this video so you can have time to answer this on your own and then when you're done you can just come back and then play so you can check your answer. We'll show you later. Okay? So God bless you for your answer. [Music] [Applause] [Music] [Music] [Applause] [Music] Okay, I hope you are now ready to check your answers. I'm now going to show you the correct answer and then please check your answers. This is the answer to part A and part B. Okay, the third part of our lesson is all about multiplication of binomials using three methods. The first is distributive property, the foil method, and the vertical form. So, I will give examples of these three methods. And then after those examples, you can practice on your own because I will also provide an activity so that you can better master our lesson. Okay. So let's now begin with the first examples. Okay. So we have distributive property method. How do we multiply binomial to another binomial? So two two terms and then two terms too. So using the distributive property, we'll just distribute our terms one by one. So let's have the first term of our first binomial. So we just multiply x by the first term of the second binomial which is x. So x x. After that we will distribute it to the second term of our second binomial which is 3. So x * 3. And then next our second term for our first monomial or I mean binomial we will multiply it by the first term of the second binomial. So 8 times. And then last, we will distribute 8 by 3. So it will be 8 * 3. Now, let us operate to simplify this. So, we have x * x, we have x s * 3, we have 3x. And then 8 * x we have 8x. And then lastly, 8 * 3 is 24. Now, can we see some similar uh terms here? Yes, there is . So, we can combine those similar terms. We have here 3x and then 8x. We can combine this. It will be 11x. Therefore, our final answer is x² + 11x + 24. Okay. Another method that we can use is what we call the foil method. So using the same example, how do we use this foil method? So first, what is foil? Well, foil is just an acronym, okay? It stands for something. What is that f? Okay, f stands for the first terms. Okay? So, in our binomials, which ones are our first terms or the previous terms? So, we have here x. So, our first binomial, x is the first. Here too, in the second binomial, our first term is x. What is the O? O stands for the outer terms. Or the terms on the side. For example, this is on our side here on our left side is x. And then this is the one on the side, which is the 3. Next, I stands for if there is an outer term, there is also an inner term. Okay. So our inner terms are 8 and x. They are the ones in the middle. Okay. And then the L is if there is a first, there is also a last. Okay. Last term. So we have here 8 and then 3. So what do we do with these? We just have to multiply them. So what we're just multiplying is if this f what we're just multiplying is the first terms. So we have x * x. So x x the answer is x s. Next one. Or we will multiply now the outer terms which is x and then 3. So x * 3 is 3x. And then next i we will multiply the inner terms which is x 8 and x. Okay? So 8 * x is 8x. And then finally, we multiply our last terms which are 8 and 3. We will get 24. Okay. Can we combine similar terms here? Yes, we can combine the outer terms and inner terms of their product. So we have here 3x + 8x, we will get 11x. Therefore, our answer is the same. So we have x² + 11x + 24 using the foil method. Okay, let's have another example. But this time you can do this on your own. You can do this. You can just pause this video if you want to do it first and then just play it if you want to check your answer. Okay? So let's begin. We have here x - 4 * x - 9. Okay? How is that? Let's start with the first terms. So we still have nothing here . X * x we will get x s. Next is the O which is the outer terms. So x times, including sin. Okay? So let's not leave the sign here. We're not just multiplying 9 by x, we're also including the negative sign. So x * -9 is -9x. Next one is our inner terms. So we have - 4 including the uh sign here or the operation. x we get -4x and then last our last terms -4 * -9 our answer is 36 okay let's combine similar terms. Our similar terms here we have -9x and -4x. Therefore it will be -13x. Our answer is x² - 13x + 36. Okay. Let's discuss our last method here which is the vertical method. Using the same example, what we can do is to line it up just like in multiplying numbers. Okay? How? So we have here x + 8 and then x + 3. So here in the vertical method, let's first do ah x here at the bottom. Okay? Our second binomial. We just multiply it by its height which is x. That's why it's called vertical. Okay, let's go to the top. So, x * x we will get x². After that, we are now going to multiply x by 8. Okay, diagonally. So, x * 8 we get 8x. Now, after this, let's follow 3. Okay, so 3 * x we get 3x. We'll just confess him here to 8x. That's his similar term. And then after multiplying 3 and x, we will also multiply it by 8. So where do we convert? He is here. So it will be 3 * 8, we will get 24 or + 24. Next we will add these two lines. We will add them. Let's just line up similar terms like this. And then we can now bring down this x s. Bring down x s we will add 8x and 3x so it will be 11x and then we will bring down 24 so it will be + 24 answer is x² + 11x + 24 that's the vertical method another example this you can also do it on your own you can just pause this video or if you want you can follow along with this example okay let's start. Let's just write it here x - 5 and then below is the second binomial x + 7. Next, let's take the first term of the second binomial first. He's upstairs. X * x we will get x². Next one. X-5. That includes the operation. X * -5 is -5x. Next is 7. 7 * x we get 7x. Let's line it up with -5x here. 7 * 5 we get -35. And then we will now add this. Let's bring down x². And then we will add -5x and 7x. We will get + 2x and then we will bring down - 35. Okay. Therefore, our final answer is x² + 2x - 35. Okay. So out of our three methods, which one do you like the most? Which one of you is the easiest? So do we need to do the three methods? No, no. You just have to choose which method is easier for you. Are you going to use the distributive, the foil method, or the vertical method? The answer will be the same. Okay. So, if you are ready, let's have this activity so you can practice our multiplication of binomial to another binomial. So I'm going to give you time to do this. You can use any method you like, any method that is easier for you. Okay? Because the answer will be the same whether you use any method among the three examples that I have given. Okay? So, God bless you for your answer. Okay, I hope you are now ready to check your own answer. I'll show you our answers and then you can check if they're the same as what you did. Okay? So regardless of what method you have used, that is what the final answer will be. Okay? So I hope you got it right. I hope you understood something from our video tutorial. Our tutorial today is a bit long, but I hope we understood. Okay. So again, thank you for listening and for our next lesson, please stay tuned and see you again in the next video. If you learned something from this video, please click the subscribe button to support this channel so I can help you more with your next math lessons. Always remember that math is for everyone. And I, teacher Van, say that here in Math, only the mind thinks. God bless and see you on the next video. [Music]