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.

(a−b)(a²+ab+b²)

Accepted Answer

Hey, everyone in this problem, we're asked to multiply the following polynomials and simplify the resulting product. The polynomials were given, we have the first polynomial X -2 V and that's multiplied by the second polynomial X squared plus two X V plus four V squared. We're given four answer traces. Option A is 12 X cubed minus V cubed. Option B is A X cubed minus V cubed. Option C is X cubed minus 12 V cubed. And option D is X cubed minus eight V cubed. Let's start by rewriting the polynomials. We've been given, we have X -2 V and that's multiplied by the polynomial X squared plus two X V plus four V squared. Now, when we're given two polynomials like this and we want to multiply them together, what we wanna do is take each term of our first polynomial and multiply it by every term in the second polynomial. So we're gonna start by taking the first term of our first polynomial X and we're gonna multiply it to the first term of the second polynomial X squared. Then we're gonna multiply it to the second term of the second polynomial to X V. And then we're gonna multiply it to the last term of the second polynomial four V squared. And then we're gonna do the same with the second term. So we're gonna take negative two V, we're gonna multiply it to the first term of the second polynomial X squared. And we're gonna multiply it to the second term of the second polynomial. And finally to the third term of the third or of the second polynomial. And that is gonna be how we multiply these polynomials together. And when we do that, what we're gonna get is the following, we have X multiplied by X squared. OK. So that's the first term of the first polynomial, first term of the second polynomial. And then we're gonna add, we're sticking with the first term of the first ex uh polynomial multiplied by the second term of the second polynomial, two X V. And then we go to the next term, add X multiplied by four V squared. Now we're done with the first term of the first polynomial we can move on to the second term. So now we're gonna add, we're gonna multiply and don't miss this sign. OK? We need to multiply by negative two V and we're multiplying to the first term of the second equation or sorry, the second polynomial X squared can we take negative two V multiply it to the second term? Two X V and we add one more term here negative to V. The second term of the first polynomial multiplied by the last term of the second polynomial for V squared. And that is our expanded expression and we've done the multiplication. Now, now it's just a matter of simplifying in the first term, we have X multiplied by X square. When we multiply two things together that have the same base recall, we add the exponents. So this becomes X to the exponent one plus two. Our next term we have x multiplied by two XV. Well, the only constant coefficient we have is two. So we have two, we have multiplied by acts. That's gonna be X to the exponent one plus one multiplied by V. The next term four V squared multiplied by X, we're gonna write all of our terms so that the X shows up before the B OK. So the X first, then the V just to keep it consistent, it's gonna be easier to simplify the expression and not mix up terms when they're all written consistently. So we're gonna write this as four X V squared. We have a negative two B multiplied by X squared. This is gonna be negative two X squared, V negative two V multiplied by two X V. So we have negative two, multiplied by two, gives us negative four. We have an X and then we have V multiplied by V. So we get V to the exponent one plus one. In our final term. We have negative two multiplied by four. So the constant is negative eight and then we have V multiplied by V squared which gives us V to the exponent one plus two. OK. We can simplify the exponents here. So we have X to the exponent one plus two gives us X cubed. We have plus two X to the exponent one plus one multiplied by V gives us two X squared V plus four X V squared minus two X squared V minus four X V to the exponent one plus one, V squared minus eight V to the exponent one plus two gives us V cued. So all of our exponents are simplified. Now, now we can simplify the expression. One last step, notice that we have a term plus two X squared V and we have a negative two X squared V, those two terms are going to add to zero. And similarly, we have plus four X V squared minus four X V squared. Those two terms are also going to add to zero. And so we're only left with two terms X cubed minus eight V cubed. And that is a simplified expression of the multiplication of these two polynomials. Comparing to our answer choices, we found the expression was X cubed minus eight V cubed which corresponds with answer choice. D. 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. (a−b)(a²+ab+b²)

Key Concepts

Factoring and Expanding Polynomials

The Distributive Property

Special Products

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