In Exercises 133–136, factor each polynomial completely. Assume t...

Question

In Exercises 133–136, factor each polynomial completely. Assume that any variable exponents represent whole numbers.

36x²ⁿ − y²ⁿ

Accepted Answer

Welcome back. I am so glad you're here. We are asked to perform complete factorization of the following polynomial expression. Our expression is 64 N raised to the two P minus M raised to the two P. And our answer choices are answer choice. A open parentheses, eight N raised to the P plus M close parentheses multiplied by open parentheses. Eight N raise to the P minus M close parentheses. Answer choice, B open parentheses, eight N squared plus M squared. Close parentheses raised to the P. Answer choice C open parentheses, eight N raised to the P plus M. Raise to the P closed parentheses squared and answer choice. D open parentheses. Eight N raised to the P plus M. Raise to the P closed parentheses multiplied by open parentheses. Eight N raised to the P minus M. Raise to the P close parentheses. All right. The first step when we are factoring is to look at our terms and see if they have any common factors that we can pull out and looking at the coefficients. 64 and one don't have any common factors and looking at our variables N and M are not common variables. So we cannot factor anything out in front. The next thing we do because we only have two terms here is we discern whether we might be dealing with a situation with a difference of squares, a difference of cubes or a sum of cubes. And so let's take a look at what we've got is eight squared N raised to the two P. That's essentially N raised to the P squared and M raised to the two P, that's M raised to the P also squared. And we know that our variables are raised to the P and then squared because if we were to take the P raised to the power of two, we would be multiplying P times two which gets us two P, which is what we started with for both of those N and M variables. These are being subtracted from each other. Everything here is a square. So we have a difference of squares. We recall from previous lessons that when we have a difference of squares that could be written, expressed as open parentheses. A squared minus B squared, closed parentheses. And this is factored as open parentheses. A plus B, closed parentheses, multiplied by open parentheses. A minus B closed parentheses. Now, all we need to do is identify our A and B terms and then we can fill them into the factored difference of squares. And then we'll have factored our original polynomial expression. So first we want to identify our A term. The A term is the first thing being squared here. That's going to be this eight N raised to the P. So A is equal to eight N raised to the P. Our B term is the second thing being squared, the one being subtracted. So our B term is going to be M raised to the P. So now that we've identified these, anywhere in our factored difference of squares where we have an A, we'll be replacing that with eight N raised to the P. And anywhere we have A B, we'll be replacing the B with M raised to the P. So let's go ahead and do that. So we'll have open parentheses instead of a, we'll have eight N raised to the P and then bring down plus and then instead of B, we have M raised to the P closed parentheses multiplied by open parentheses. Instead of a, we'll have eight N raised to the P and then bring down minus. And instead of B, we have M raised to the P closed parentheses. And this is our polynomial expression factored. We look at our answer choices and this matches with answer choice B D. Excuse me. D well done. We'll catch you on the next one.

In Exercises 133–136, factor each polynomial completely. Assume that any variable exponents represent whole numbers. 36x²ⁿ − y²ⁿ

Key Concepts

Factoring Polynomials

Difference of Squares

Exponents and Variables

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