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.

2x² + 9x + 7

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. Our trinomial is two A squared plus 17 A plus 35. Our answer choices are answer choice. A open parentheses, two, A plus seven closed parentheses multiplied by open parentheses. A plus five closed parentheses. Answer choice, B open parentheses, two, A plus five closed parentheses multiplied by open parentheses. A plus seven closed parentheses. Answer choice C Open Parentheses. Two, a plus 35 closed parentheses multiplied by open parentheses. A plus one closed parentheses. Enhanced choice d the given trinomial is prime. The first step in factoring a train, no meal is that we look at all three of our terms and see if there are any common factors that we can factor out between 2 17 and 35. Those do not share any common factors and not all three of our terms have an a variable. So we can't factor out a variable either. And therefore the next step is that we set up two open sets of parentheses and we're essentially doing the reverse of the foil process, we ask ourselves what multiplied by what equals two A squared. Those will be our firsts for that A squared. That's going to be a multiplied by A, for the two. That could be two times one or one times two. Uh, order does not matter for our first. So to keep it consistent with our answer choices, for the first set of parentheses, I'll have two A and for the second set of parentheses, the first term will just be a, We've got our first established. So now we look at our lasts, we ask ourselves what multiplied by what equals a positive 35. And we've got options for 35, we could have one multiplied by 35. We could have negative one multiplied by negative 35 order. Does matter for our last. So we could have 35 multiplied by one or negative 35, multiplied by negative one. We could have five multiplied by seven, negative five, multiplied by negative 77, multiplied by five or negative seven, multiplied by negative five. So a lot of different answer choices for our last. How do we choose? Well, what we do is we now look at our outsides which are going to be the first, from the first set of parentheses, multiplied by the last, from the second set of parentheses and add those to our insides, which will be the last from the first set of parentheses multiplied by the first from the second set of parentheses. And those need to equal our second term in our trinomial, which is this positive 17 a and a positive 17 A tells us that our lasts will also need to be positive. So we can eliminate all of those negative factors. We can cross off negative one multiplied by negative 35 negative multiplied by negative one, negative five, multiplied by negative seven and negative seven multiplied by negative five. All right, we've got four of them crossed off. We have four different sets of factors left. But instead of just going right in and plugging them all in, we have multiple choice selections here and we can just start off with answer choice A and plug in those lasts what they've got in answer choice A are a positive seven and a positive five. So let's plug those in. Our first set of parentheses will read two A plus seven and our second set of parentheses will read a plus five. So now let's foil those out and see if they work for our first. We have two, a multiplied by A which is two A squared. For our outsides. We have two, a multiplied by five which is plus 10 A for our insides, we have seven multiplied by A which is plus seven A. And for our last, we have seven multiplied by five which is plus bringing down our first two A squared. Combining our outside and inside like terms in the middle A plus seven A is plus 17 A and bringing down our last plus 35 this does match what we started with. So we have factored this correctly and we look at our answer choices. Open parentheses, two A plus seven, closed parentheses multiplied by open parentheses. A plus five closed parentheses matches with answer choice. A 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. 2x² + 9x + 7

Concetti chiave

Factoring Trinomials

FOIL Method

Prime Trinomials

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