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.

5y² + 33y − 14

Accepted Answer

Welcome back. I am so glad you're here. We are asked to factor the following Trina meal if possible to verify if the answer is correct by using the foil method. And our Trina meal is nine M squared plus 70 M minus 16. Our answer choices, our answer choice. A parentheses, nine M minus four, end parentheses times new parentheses. M plus four, end parentheses. Answer choice B parentheses, nine M minus eight, end parentheses, times new parentheses, M plus two and parentheses. Answer choice C nine M excuse me, parentheses. Nine M minus two and parentheses times new parentheses, M plus eight and parentheses and answer choice D is that the given try, no meal is prime or can't be factored. All right. So the first step in factoring is we're going to look at all three of these terms and ask ourselves if they have any common factors that we can factor out. And 970 and 16 do not share any common factors and not all three of them have an M. So we can't factor out a variable either. Therefore, our next step is we're going to set up two sets of parentheses and we're going to do essentially the reverse of the foil process. We're going to ask ourselves what times what equals nine M squared. Those will be our firsts as far as the M squared is concerned, that's going to be an M times and M. But for that nine, we've got options, we could have three times three, we could have nine times one. An order doesn't matter for the first and the first are typically positive. So we can leave it with those two options because we've got options. We don't have anything solidified. We can move on to try to help narrow down our choices. And we're going to now ask ourselves what times what equals a negative 16. Those will be our lasts. And for a negative 16, we've got options negative 16 Because order does matter could be a one times a negative 16 or it could be a negative one times a positive 16 or it could be a negative 16 times a positive one or a positive 16 times a negative one. We also have, we could have two times eight, excuse me, one of those will be negative two times negative eight or negative, two times a positive eight or we can have the positive eight times the negative to or the negative eight times the positive two or we could have a positive four times a negative four or a negative four times positive four. That's a lot of options. Well, how do we narrow this down. How do we figure out which ones are the correct ones? Well, when we put them in, we will then multiply our outsides. The outsides are the first from the first set of parentheses times the last from the second set of parentheses. And then we'll add that to our insides. The insides are the last from the first set of parentheses times the first from the second set of parentheses. When they add up, they will need to equal a positive 70 M. The middle term from our try no meal. So figuring out which will equal a positive 70 M. Well, first, you could say 70 is a pretty big number. So we might need some bigger variables to be multiplying to get that. And so you could start off throwing in some of the numbers that have bigger variables. However, we also note that this is a multiple choice problem and they've got three possible solutions for us to try instead of all of these different possibilities. And for answer choices A B and C, the firsts that they've chosen for all three are nine times one. So we can go ahead and fill in a nine times one for our first. So we'll have nine M and one M R just M for our first. And now let's just start with answer choice A for the last they have negative four and positive for, I don't know if those are going to be big enough numbers, but we can go ahead and try them. So we'll have nine M minus four times M plus four. Now, multiplying our first, now we'll foil it to check first. R nine M times M that's nine M squared, which is by design, we knew that would work. Now, for our outsides, we have nine M times four. That's plus 36 M. For our insides, we have negative four times M. That's going to be a minus four M and then our lasts will have negative four times a positive for that's minus 16, which is by design. We knew that would work. Now, combining those like terms in the middle are outsides and insides. We have 36 M minus four M that equals positive 32 M. Nope, not big enough, not quite to our 70 M. So these don't work. We can eliminate answer choice A and take our negative for positive for off the table for our lasts. So now let's try the next thing and this time, we don't have to re multiply our firsts and our lasts. We know that those will work. But we can now try for our last, let's try from answer choice B negative eight times two. So we've got nine M minus eight times M plus two and multiplying our outsides, we have nine M times two, which will be A plus 18 M. These are not going to get very big and then for our insides. We have negative eight times M which is minus eight M combining like terms here 18 M minus eight M equals M. That's even smaller than the last one. That doesn't work. We can eliminate answer choice B and we can eliminate our options of negative eight and positive two from our lasts. Okay. Let's try what they've got for answer choice, C for answer choice C we've got negative two and positive eight for our lasts. So we'll try negative to positive eight. So that's nine M minus two times M plus eight. Multiplying our outsides. That's nine M times eight. That's plus 72 M. Oh, that looks good. And then our insides negative two times M that's minus two M combining like terms. We have 72 M minus two M. That's 70 M. Yes, that's what we wanted. Let's bring down our nine M squared plus our M minus the 16. This matches we have factored correctly. And so we want nine M minus two times M plus eight, which is answer choice C 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. 5y² + 33y − 14

Key Concepts

Factoring Trinomials

FOIL Method

Prime Trinomials

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