In Exercises 65–74, factor by grouping to obtain the difference o...

Question

In Exercises 65–74, factor by grouping to obtain the difference of two squares. z² − x² + 4xy − 4y²

Accepted Answer

Hello, today we're gonna be factoring the given polynomial. So what we are given is Q squared minus M squared plus six M N minus nine N squared. Now, normally, when we see a polynomial of four terms, we should always try factoring by grouping. But notice that if we try to do factoring by grouping, we cannot factor out a common factor from the first group. So factoring by grouping is not going to work in this case. So notice that we have negative M squared plus six M N minus nine N squared. This is very close to the form of a factor trinomial, but we need to factor out a negative one from the terms. So if we group up these last three terms and factor out a negative one, we can rewrite the polynomial to be Q squared minus the quantity M squared minus six M N plus nine N squared. And this trinomial is going to factor to give us the quantity M minus three N squared. So what we have is Q squared minus M minus three N squared and Q squared is a perfect square because it's Q times itself two times and M minus three N is a perfect square because it's the quantity M minus three N multiplied by itself two times. So what we have here is a difference of squares and the difference of square states that if you have A squared minus B squared, this is going to factor to A minus B multiplied by A plus B. Here, we're going to allow A to equal to Q and we're going to allow B to equal to M minus three N. So if we plug this in to the difference of squares, we can factor this to give us A minus B which is Q minus the quantity M minus three N multiplied by A plus B which is going to be Q plus the quantity M minus three N. So on the left hand side, we're gonna go ahead and distribute the negative one into the quantity M minus three N. And this is going to change the signs of the terms leaving us with Q minus M plus three N. Now, what we're going to do is we're going to alphabet alphabetically order each term in both of the groups. And that is going to give us negative M plus three N plus Q multiplied by the quantity M minus three N plus Q. And we want the leading coefficients to be positive in each group. So we're going to need to factor out a negative one from the first group. And doing so is going to leave us with negative one times the quantity M minus three N minus Q multiplied by M minus three N plus Q. And this is going to be the completely factored version of the given polynomial. And with that being said, the answer to this problem is going to be B 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 65–74, factor by grouping to obtain the difference of two squares. z² − x² + 4xy − 4y²

Key Concepts

Factoring by Grouping

Difference of Squares

Polynomial Expressions

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