Factor each polynomial. See Examples 5 and 6. (b+3)^3-27

Question

Accepted Answer

Hello, today we're going to be factoring the given expression. So what we are given is the quantity P plus six cubed minus eight. Now there's one change that we can make to this expression before we go ahead and factor it. And that is for eight because eight is the same thing as two cubed. So if we replace eight with two cubed, what we get is the quantity P plus six cubed minus minus two cubed. And essentially what we have here is a difference of cube terms which has a special factor when you have a difference of two cubed terms, which is X cubed minus Y cubed. This is going to be able to factor two X minus Y times the quantity X squared plus X times Y plus Y squared. Here, we're going to let X equal to the quantity P plus six and we're going to let Y equal to two. So if we take these values of X and Y and plug it into the special factor for a difference of cubes, what we end up with is X minus Y which is P plus six minus two times the quantity X squared. Which is P plus six squared plus X times Y, which is going to be two times P plus six plus Y squared, which is going to be two squared. So there's a bit of simplification that we can go ahead and do first things first, we can go ahead and add six with negative two and that's going to give us positive four. Next, we can distribute two into P plus six and we know that two square is going to evaluate to give us four. So making these simplifications is going to give us P plus four times the quantity P plus six squared plus two times P which is two P plus two times six, which is 12 plus four. Now, we can go ahead and combine like terms here by idea 12 and four together, which is going to give us 16. So the next thing we're going to want to do is to go ahead and foil out P plus six squared and P plus six squared is the same thing as saying P plus six times P plus six. In order to foil this, we're gonna multiply P with P plus six and we're going to multiply six with P plus six. And this is going to give us P times P which is P squared, P times six, which is six P, six times P which is another six P and six times six, which is going to give us 36. Next we can go ahead and combine like terms and we're gonna go ahead and combine the 26 piece together to give us P squared plus 12 P plus 36. And we can go ahead and replace P plus six squared with this quantity doing, this is going to give us P plus four times P squared plus 12 P plus plus two P plus 16. Once again, we can go ahead and combine like terms, we can add 12 P with two P and we can add 36 with 16. And that's going to give us P plus four times the quantity P squared plus 14 P plus 52. And this is going to be the completely factored version of the original expression. And with that being said, the answer to this problem is going to be be. So I hope this video helps you in understanding how to factor this expression. And I'll go ahead and see you all in the next video.

Factor each polynomial. See Examples 5 and 6. (b+3)^3-27

Key Concepts

Factoring Polynomials

Difference of Cubes

Binomial Expansion

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