In Exercises 45–68, use the method of your choice to factor each ...

Question

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.6x² + 19x + 15

Accepted Answer

Welcome back. I am so glad you're here. We are asked to factor the following trinomial if possible and to verify if the answer is correct by using the foil method. All right, we are given the trinomial six P squared plus 25 P plus 14. Our answer choices are answer choice. A open parentheses. Two P plus one closed parentheses multiplied by open parent. C three P plus 14 closed parentheses, answer trace B open parentheses, two P plus seven closed parentheses multiplied by open parentheses. Three P plus two closed parentheses. Answer choice C two multiplied by open parentheses. P plus one closed parentheses multiplied by open parentheses. Three P plus seven closed parentheses. In answer choice D the given trinomial is prime or it can't be factored. All right. The first step in factoring a trinomial is we look at all three of our terms and ask ourselves if there are any common factors that we can factor out. So looking at the coefficients in the constant terms. 6 25 and 14, those do not share any common factors and not all three of the terms have a P variable. So we can't factor out any variables either. So there is no factors that we can factor out in front. Now, we'll set up two open sets of parentheses and we'll do essentially the reverse of the foil process. We begin by asking ourselves what multiplied by what equals six P squared. Those will be our firsts. And as far as that P squared is concerned, that's going to be P multiplied by P. But for that six, we've got options. We could have one multiplied by six. We could have two multiplied by three order, doesn't matter for the first. So I'm going to leave it as those two options. Now for our next clue because we can't finalize anything, finalize anything with our first. We look at our lasts, We ask ourselves what multiplied by what equals a positive 14. And again, we've got options for 14. That could be one multiplied by 14. It could also be negative one multiplied by -14. Order does matter for the last. So we could have 14 multiplied by one or negative 14, multiplied by negative one. We could also have two multiplied by seven, negative two, multiplied by negative 77, multiplied by two or negative seven, multiplied by negative two. A lot of different options for our last and again, nothing solidified. So the next place we look for clues is with our outsides, which will be the first from the first set of parentheses multiplied by the last from the Second set of parentheses added to our insides, which is the last from the first set of parentheses multiplied by the first, from the second set of parentheses added together the outsides and insides need to equal a positive 25 p. Well, we've got a clue with that positive 25 P because it's positive, we can rule out all of the negative factors for 14. So we can rule out negative one multiplied by negative 14, negative 14, multiplied by negative one, negative seven, multiplied by negative, negative two, multiplied by negative seven and negative seven multiplied by negative two. So that narrows it down a little bit. And typically we would just start filling in factors for our first and last and see if combining the outsides and insides gives us that positive 25 p. However, this is multiple choice and we can look at our multiple choice solutions to see which factors we could fill in to narrow it down a bit. So we can start with answer choice, a we see that for the first they have two and three. So we can go ahead and try those for our first two P and three P And then for the lasts for this answer choice a they have one and 14. So we can give those a try. So our first set of parentheses will read two P plus one, closed parentheses multiplied by open parentheses. Three P plus 14, closed parentheses. And now we can foil this to check. We take our first two P multiplied by three P that gives us six P squared by design. We knew that one would work for our outsides. We have two P multiplied by 14. That's plus 28 P. Our insides, we have a positive one multiplied by three P. That's going to be a plus three P. And for our last, we have a positive one multiplied by a positive 14, which will give us plus 14. Now, we can bring down the first six P squared. Combine our outsides and insides the like terms here in the middle 28 P plus three P will give us a positive P and then we can bring down the last plus 14. Well, positive 31 P does not match our positive 25 P that we were going for. So this combination does not work. Sorry for one and 14 coupled with two and three, you don't work. So let's keep the first and the last we know that those multiply correctly and we'll fill in some different answer choices. We know that answer choice B has a two P and a three P again for the first. So we can keep those, but it has a different set of lasts. So instead of one and 14, we will try seven and 2. All right. So the first set of parentheses is open parentheses, two P plus seven, closed parentheses multiplied by open parentheses, three P plus two, closed parentheses, we know our firsts and our lasts will work. So let's do the outsides. We have two P multiplied by two, which is going to be plus four P and then we have seven multiplied by three P, which will be plus 21 P, combining our outsides and insides in the middle. Here, we'll have four P plus 21 P, which is A plus 25 P and that does match what we started with. So we know we have factored this correctly. So that's open parentheses. Two P plus seven, closed parentheses, multiplied by open parentheses. Three P plus two closed parentheses, which is answer choice B well done. We'll catch you on the next one.

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.6x² + 19x + 15

Key Concepts

Factoring Trinomials

FOIL Method

Prime Trinomials

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