In Exercises 69–80, factor completely.

(x − y)⁴ − 4(x − y)²

Question

In Exercises 69–80, factor completely.

(x − y)⁴ − 4(x − y)²

Accepted Answer

Hello, today we're gonna be completely factoring the given expression. So what we are given is M minus N raised to the fourth power minus 16 times the quantity M minus N squared. Now we can quickly rewrite the first term M minus N to the fourth to be M minus N squared times the quantity M minus N squared. And the reason why we want to rewrite it like that is because notice that in the difference, we have at least one instance of M minus N squared in both of the terms. So we can go ahead and factor out M minus N squared from the expression and we can rewrite the expression as M minus N squared times the quantity M minus N squared minus 16. Now, 16 itself is a perfect squared as 16 is equal to four squared. So we can go ahead and rewrite 16 as the four squared. And what we have here is a difference of square terms. If we want to factor a difference of squares. The equation states that if we have the quantity in the form of A squared minus B squared, this can be factored to A plus B times the quantity A minus B. In this quantity here, we're going to let A equal to M minus N and we're going to let B equal to four and plugging this into the difference of squares is going to give us A plus B which is going to be M minus N plus four times the quantity A minus B, which is going to be M minus N minus four. And we want to be careful because we also factored out an M minus N squared in the beginning of the problem. So we're going to include that value at the end of our factors, which is going to be M minus N squared. So what we have here is M minus N plus four times M minus N minus four times M minus N squared. And this is going to be the completely factored version of the given expression. And with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 69–80, factor completely. (x − y)⁴ − 4(x − y)²

Key Concepts

Factoring Polynomials

Difference of Squares

Substitution Method

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