Factor each polynomial completely. See Example 6.x² - 2x - 15

Question

Accepted Answer

Welcome back. I am so glad you're here. We're asked to rewrite the following expression by factoring it completely. Our expression is P squared plus six P minus 27. Our answer choices are answer choice. A, the quantity of P minus six multiplied by the quantity of P plus three answer choice B, the quantity of P plus three multiplied by the quantity of P minus nine answer choice C, the quantity of P plus six multiplied by the quantity of P plus three and answer choice D, the quantity of P minus three multiplied by the quantity of P plus nine. All right. So looking at our expression, we know first of all that there's just an invisible unwritten one in front of our P squared and that our third term of our trinomial doesn't have any variable. It doesn't have A P. So there's nothing that we're able to factor out from all three of those terms. So now for factoring, we can begin with the reverse of the foil process, setting up two open sets of parentheses, we begin by asking ourselves what multiplied by what gives us P squared. Those are our firsts. And for P squared. That's just going to be P multiplied by P. We next ask ourselves what multiplied by what gives us a negative 27. Those are going to be our lasts. Now for negative 27 we have a lot of options. I'll spell some of them out here for negative 27. We could have a negative one multiplied by a positive 27 or we could have a positive one multiplied by a negative 27. We could also have a negative three multiplied by a positive nine or a positive three multiplied by a negative nine. We could also have all of those in the reverse order. But stopping with those for the moment because the coefficient of our P squared term is one, we can actually do this a little bit faster. We have some good clues here because we know that our outsides multiplied together, added with our insides multiplied together need to add up to a positive six P. So our factors of negative 27. This is because the P squared term has a coefficient of one. The factors of our negative 27 need to add up to a positive six. Well, we can eliminate a lot of these. We know that negative one and positive 27 don't add up to six, same with one and negative 27. Now we know that negative three and positive nine do add up to six, but three and negative nine do not So we've got our factors and we can fill these in. So we have a P minus three in the first set of parentheses. And the second set of parentheses is P plus nine and we can always foil those out just to check, we'll do it real quick. So our first P multiplied by P or P is P squared, our outsides P multiplied by nine is plus nine P insides negative three, multiplied by P is negative three P and lasts negative three, multiplied by nine is negative 27. Combining like terms in the middle nine P minus three P is that six P. And this does work. So we've got our answer. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Factor each polynomial completely. See Example 6.x² - 2x - 15

Key Concepts

Factoring Polynomials

Quadratic Formula

Zero Product Property

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