Factor: x² + 4x + 4 − 9y². (Section 5.6, Example 4)

Question

Accepted Answer

Hello, today we're gonna be factoring the given expression. So what we are given is M squared plus 10 M plus 25 minus 49 N squared. Now, normally when you see a polynomial of four terms, you wanna go ahead and try factoring by grouping first. But notice that when we make the groups for factoring by grouping, we can factor out an M from the first group, but there are no common multiples or common terms that we can factor out from the second group. So that means that factoring by grouping isn't going to work in this scenario. Now, one thing to notice about the problem if we go ahead and group up the first three terms together M squared plus 10 M plus is a perfect square trinomial. See, in order to factor this, we need to think what are two numbers that we can multiply together to give us 25, but can add up to give us 10. While those two numbers are going to be five and 55 times five, gives us 25 5 plus five gives us 10. So M squared plus 10 M plus 25 factors to be M plus five times M plus five. And this itself can be simplified and rewritten as M plus five squared. Now, this is just a factored version of the first three terms. We still need to include that last term given to us, which is going to be -49 and squared. Now native 49 N squared is also a perfect square, see seven N times seven N multiplies to give us 49 N squared. So we can go ahead and rewrite 49 N squared as seven N times seven N or as seven N quantity squared. So the complete expression can be rewritten as M plus five squared minus seven N squared. And what we have here is a difference of squares. The way to factor a difference of square states that if you have the quantity A squared minus B squared, this is going to factor to A plus B times A minus B. And in this case here, we're going to let a equal M plus five and we're going to let B equal to seven N. So plugging this in to the difference of squares is going to give us A plus B which is going to be M plus five plus seven N times A minus B which is going to be M plus five minus seven N. 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 c 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.

Factor: x² + 4x + 4 − 9y². (Section 5.6, Example 4)

Key Concepts

Factoring Quadratic Expressions

Difference of Squares

Binomial Products

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