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.8y² + 10y + 3

Accepted Answer

Welcome back. I am so glad you're here. We are asked to factor the following Trina meal if possible and to verify if the answer is correct by using the foil method. The tri no meal we're given is 15 K squared plus 52 K plus 45. And our answer choices are answer choice. A parentheses, three K plus five, end parentheses times new parentheses, five K plus nine, end parentheses. Answer choice, B, five times parentheses, K plus one end parentheses, times new parentheses, K plus nine and parentheses, answer choice C parentheses, 15 K plus nine and parentheses times new parentheses K plus five and parentheses and answer choice. D the given try no meal is prime. Alright. Well, the first step in factoring, try no meals is we're going to take a look at all three of these terms and see if they have any common factors. And 1552 and 45 do not share any common factors that we can factor out. And not all three terms have a variable. So we can't factor out a variable either. So the next step we will do is set up two sets of parentheses and we're basically going to do the reverse of the foil process. So we ask ourselves what times what equals K squared. Those are going to be our firsts. And for 15 k squared for the K squared, that's going to be A K times A K. But for the 15, we've got options for 15, we could have one times 15, we could have three times five and order doesn't matter for the firsts and first. So typically positive. So we can leave it with just those two options. But because we have options We need to pause on the firsts and now look at something else. We are going to look at the lasts. We ask ourselves what times what equals a positive 45, those will be our lasts. And for 45 again, we've got options. But this time More options, we could have one times 45. We could also have negative one times negative 45. An order does matter for the last. So we could also have 45 times one or negative, 45 times negative one. We could have three times 15, negative, three times negative, 15, 15 times three or negative, 15 times negative three. We could also have five times nine, negative, five times negative nine or nine times five or negative nine times negative five. So a lot of options for that 45 where do we even start? Well, how do we know what we're going? To get, we're going to take our outsides, which is the first from the first set of parentheses times the last from the second set of parentheses. And then we're going to add those to our insides, which is the last from the first set of parentheses times the first, from the second set of parentheses, we add our outsides and insides together and they need to equal positive 52 K, this middle term from our try no meal and because they need to add up to be a positive 52 K, we can eliminate all of the negatives from our last options from the factors of 45. So we can cross out negative one times negative 45 negative, 45 times negative one, negative three times negative 15, negative, 15 times negative three, negative five times negative nine and negative nine times negative five. Alright, that narrowed it down a little bit. But what else can we do? We just start plugging in all of these first and last and see what adds up to K. Well, normally that would be what we'd have to do however, and this is multiple choice and we've got three answer choices here that have numbers and variables. So let's take a look at those and see what they've done here. We see that answer choice be factored out of five and we know we can't factor out of five from all three of these terms. So we can eliminate answer choice B And now looking at answer choices a and see one of them used for the first three and five and the other one used 15 and one. So let's go ahead and start from the top. Let's try answer choice A which used three and five for the first. So they will be three K and five K in our first. And then the last that they chose are five and nine. So we can have three K plus five times five K plus nine attempting to use these for our firsts and our lasts. So now we do the foil process taking our first, which is three K times five K that gives us K squared, which we knew that was by design. Now we take our outsides three K times nine. That is plus 27 K. Now our insides five times five K, that's plus 25 K and now our lasts which we know should work five times nine. That's plus 45. Now adding up the like terms in the middle for our outsides and insides. We have 27 K plus 25 K and that comes out to 52 K. That's what we were looking for. We can bring down the first. So 15 K squared plus that 52 K that we were looking for and then plus our last. So plus 45 this matches so we have correctly factor this out. We are looking for three K plus five times five K plus nine which matches with answer choice a while 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.8y² + 10y + 3

Key Concepts

Factoring Trinomials

FOIL Method

Prime Trinomials

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