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.

4x² − 27x + 18

Accepted Answer

Welcome back. I am so glad you're here. We are asked to factor the following, try no meal if possible. And to verify that the answer is correct by using the foil method. The tri no meal that we're given is eight A squared minus 79 A plus 63. And our answer choices are a parentheses, eight a -9 and parentheses times new parentheses, a -7 and parentheses. Answer choice B parentheses for A minus nine and parentheses times new parentheses to a minus seven and parentheses, answer choice C is parentheses eight A minus seven and parentheses times new parentheses A minus nine and parentheses, an answer choice D is that the given Trina meal is prime or in other words, we can't factor it. Alright. When factoring, try no meals, the first step, we're going to look at all three of these terms and ask ourselves, do they share any common factors that we can factor out in between eight, negative 79 and 63? There are no common factors there and not all three terms have an A in them. So we can't factor out any A variables either. So now the next step is we set up two sets of parentheses. We're basically going to do the reverse of the foil method here. So we're going to ask ourselves what times what gives us eight A squared and the, what time's the, what those are our firsts? Well, with the A square that's going to be in a times and a, but with the eight, we've got possibilities here, we could have an eight times a one. We could have a two times of four. And order does not matter for our firsts. So we can just leave it at that. Since we have options, we don't get to just plug anything in right away. And now we turn our attention to the lasts And we say what times what gives us a positive 63. And again, we've got choices and this time order does matter. So we've got a lot of choices. We could have a positive nine times a positive seven. I said the opposite of what I wrote a positive seven times a positive nine. We could have a negative seven times a negative nine. We could have a positive nine times a positive seven because order doesn't matter. Or we could have a negative nine times a negative seven. We could also have a positive 63 times a positive one. We could have a negative 63 times a negative one. We could have a positive one times a positive 63 or we could have a negative one times a negative 63. And essentially what we need to do is fill in different options for our first and our last and try different combinations until we get something where our outsides plus our insides Will give us this negative 79 a. So we could just start plugging things in, we could because this is multiple choice. Take a look at our answer choices and see different possibilities that they've given us. And they've got two of them that have eight A and one A and then one of the answer choices has a four A and a two A. So let's start with the 88 and the one A. So we plug in eight A and one A for our firsts And now for our lasts, we've got for answer choice a we've got -9 and -7. So we can plug in negative nine and negative seven testing those. And we notice that this is a good clue here. They have to add up to a negative number. So it's probably going to be a negative times a negative. We can rule out all of these positives for our different law last opportunities. So now let's do the foil process and see if it is eight A minus nine times A minus seven. So our first we know the first are going to work eight times A gives us eight A squared. Now our outsides we have eight A times negative seven. That's a negative 56 A. Now our insides negative nine times A that's a negative nine A and our last we know the last are going to work negative nine times negative seven is a positive 63. Now we just combine our outsides and insides to see if they add up to negative 79 A A negative minus nine is only going to get us as far as a negative 65 A. So that doesn't match negative 79 A, that one's not going to work. We can rule out answer choice A And so now let's try the other one Where they have eight a And one a as the firsts. And so we'll fill in some different last. They have this time. A negative seven and a negative nine. So pulling from here negative seven and negative nine, plugging those in and we know our first and our last are okay. So let's do again our outsides and then we'll do our insides. So the outside's eight A times negative nine and that is negative 72 A and the negative seven times A is minus seven A. Now combining our outsides and our insides, negative 72 A minus seven A that equals negative 79 A. So bringing down the first eight A squared and bringing down the last plus 63 this does match. So at this point, we have factored it correctly, we have eight a minus seven times a minus nine and that matches with answer choice c well done. We'll catch 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. 4x² − 27x + 18

Key Concepts

Factoring Trinomials

FOIL Method

Prime Trinomials

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