In Exercises 23–34, find each product using either a horizontal o...

Question

In Exercises 23–34, find each product using either a horizontal or a vertical format.

(xy+2)(x²y²−2xy+4)

Accepted Answer

Hey, everyone in this problem, we're asked to multiply the following polynomials and simplify the resulting product. And what we're given is the polynomial three multiplied by a multiplied by B plus one. And that polynomial is going to be multiplied by the polynomial nine multiplied by A squared multiplied by B squared minus three, multiplied by a multiplied by B plus one. We're given four answer choices a through D A is 27 multiplied by A cubed multiplied by B cubed minus nine multiplied by A squared multiplied by B squared plus three, multiplied by a multiplied by B plus one. Option B 27 multiplied by a cubed multiplied by B cubed minus nine, multiplied by A squared multiplied by B squared Plus three, multiplied by a multiplied by B -1. Option C is 27 multiplied by A cubed multiplied by B cubed plus one. And option D is 27 multiplied by a cubed multiplied by B cubed minus one. So let's get started with this problem. We're gonna rewrite the expression we're given. So we have three multiplied by a multiplied by B plus one. That's our first polynomial. And we're gonna multiply that entire polynomial by our second polynomial, nine multiplied by A squared, multiplied by B squared -3, multiplied by a multiplied by B Plus one. Now, when we're multiplying polynomials together, what we wanna do is multiply each term of the polynomial, the first polynomial by each term of the second polynomial. So we look at our first polynomial. The first term is three a B. So we wanna take three A B and we want to multiply it by the first term in the second polynomial, nine multiplied by A squared multiplied by B squared. Then we want to take it and we want to multiply it to the second term of the second polynomial is minus three, multiplied by a multiplied by B. And then we want to take it and multiply it to the third term in our polynomial plus one. Then we want to do the same for the second term. So our second term is plus one and we want to take that and multiply it to the first term of the second polynomial, the second term of the second polynomial and the third term of the, of the second polynomial. And when we do this, what are we gonna get? Well, we have that first term of the first polynomial. Three multiplied by a multiplied by b multiplied by the first term of the second polynomial. nine multiplied by A squared multiplied by B squared. Then we're gonna add the next term and the next term is gonna be that first term of our first polynomial, three multiplied by a multiplied by B. And now we're gonna multiply to the second term in the second polynomial which is negative three multiplied by a multiplied by B and don't miss those signs. Ok. It's important to include that sign in that term next. OK. We're sticking with the first term of the first polynomial. We still have to multiply it to the last term of the second polynomial. So we have three multiplied by a multiplied by B and that's gonna be multiplied by one. Now we're gonna add, and we're gonna move to the second term. OK. We've expanded the first term. We've done the first term multiplied by every term of the second polynomial. So now we're gonna do the second term of the first polynomial multiplied by all these terms. So we get one multiplied by nine, multiplied by A squared multiplied by B squared. I'm just gonna include the next two terms below because we're out of space, we add again that second term from the first polynomial, one multiplied by the second term of the second polynomial negative three multiplied by A multiplied by B. And then we're gonna add the last term of each Multiplied one x 1. All right. And that's all of the terms we have to consider. Now it's just a matter of simplifying. So our first term, we have three A B multiplied by nine A squared B squared. So we can take the coefficients, these constants three and nine and multiply them, we're gonna have 27. Now we have an A in this first expression multiplied by an A squared when we multiply with the same base recall that we add the exponents. So here we have A to the exponent one multiplied by A to the exponent two. This is gonna give us A to the exponent one plus two. And similarly, for B, we have B to the exponent one multiplied by B to the exponent two, which gives us B to the exponent one plus two, moving to our next term. In terms of the constant coefficients, we have three multiplied by negative three. So we're gonna have negative nine, we have a multiplied by A, that's gonna be A to the exponent one plus one. And similarly, for B B multiplied by B, we get B to the exponent one plus one. Moving to our next term three A B multiplied by one. It's just gonna give us three A B. Now the next three terms are a little bit simpler because the term that we're multiplying by them was just one. OK. So we have one multiplied by nine A squared B squared. This is gonna give us nine A square B squared. We have one multiplied by negative three A B. So this is gonna be negative three A B and then we have plus one multiplied by one, which gives us just one. Next step is to simplify these exponents that we've written. So our first term we have 27 We have a to the exponent 1-plus 2 and we get a cubed. We're gonna multiply that by B to the exponent one plus two, be cute. Our next term negative nine A to the exponent one plus one gives us A squared and B to the exponent one plus one gives us B squared. The rest of the terms are already simplified three A B plus nine A squared B squared minus three A B plus one. Now, our job is to simplify that entire expression. What you'll notice is we have a negative nine A squared B squared term and A plus nine A squared B squared term. So those two will add to zero, we also have three A B minus three A B, those two are gonna add to zero as well. And so all we're left with is two terms 27 multiplied by A cubed, multiplied by B cubed plus one. And that is our simplified version of the expression we were given. If we compare this to our answer choices, We found the simplified expression was 27 a cubed b cubed plus one which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 23–34, find each product using either a horizontal or a vertical format. (xy+2)(x²y²−2xy+4)

Key Concepts

Polynomial Multiplication

Horizontal and Vertical Formats

Combining Like Terms

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