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.

16y² − 6y − 27

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 that the answer is correct by using the foil method. The Trina meal we're given is 18 B squared minus 27 B minus 35. Our answer choices are answer choice. A parentheses, three B minus seven, end parentheses times new parentheses, six B plus five and parentheses answer choice B is parentheses, three B minus five, end parentheses times new parentheses, six B plus seven and parentheses answer choice C is parentheses 18 B plus seven and parentheses times new parentheses B minus five. End parentheses, an answer choice D is the Trine Amiel is prime. Alright. So what we're going to be doing to factor this, the first question we ask ourselves is, is there any greatest common factor that these three terms in the tri no meal share that we could factor out in front and between 18, negative 27 negative 35. Those don't share any common factors and B squared B and then the third one doesn't have A B. So nothing we can factor out in terms of the variables either. So the next step in factoring is that we're going to set up two sets of parentheses and we're going to try to fill these out and we're basically doing the reverse of the foil process. Here. We're going to look at the first positions first and that's what the F stands for in foil. We ask ourselves what times what equals 18 B squared. Well, as far as that B squared is concerned, that's going to be B times B. So A B for each of the two first terms, but for the 18, we've got options. So for 18, we might have three times six, we might Have two times nine, we might have one times 18. Order doesn't matter for the firsts, but we do have all three of those possibilities and we can't decide which one it is yet. So we're just going to set that to the side for a second. And now we will ask ourselves what times what will equal this third try, no meal, the negative 35. Those are our lasts in the foil process. So what times what equals negative? 35 here, we've got a lot of options because order does matter for the lasts. So we might have a negative five times a positive seven, we might have a positive five times a negative seven. We might have a negative seven times a positive five, we might have a positive seven times a negative five. We might have a negative 35 times a positive one. We might have a positive 35 times a negative one. We might Might have a negative one times a positive and we might have a positive one times a negative 35. So we have so many different combinations of possibilities here to test out what they have to do is when you take their products. So you do the multiplication and then you add them together. They are going to have to equal this middle term in our try no meal. Those are going to be our outsides and our insides. When we figure those out, the outsides and the insides. So I'll show you how this can work. We can first start testing these out. So we do have some clues here as we read our answer choices before we see that some possibilities are starting with the three in a six in the firsts. So since we've got these answer choices here, it's worthwhile to start testing A three and A six for our positions here. So we can try the firsts Um being R three With the B and the six with the B and then we can try, they've got a negative seven and a positive five for answer choice. A answer choice B has a negative five and positive seven. So let's start with answer choices. Answer choice A s, if you didn't have multiple choice, you would just have to go through using three and six for the first and then test out all of these different factors for the last, which is a lot of work. So we'll use any hints that we can get from the way the problem is written. So if we've got three B, let's try negative seven and positive five for our last. So we have minus seven and then plus five. So now we are going to foil these out. This is how we see if it is the correct one. So for our firsts, we have three B times six B and that's 18 B squared by design, we knew that part would be correct. Now, it's testing the outsides and the insides. So the outsides here three B times positive five, that's going to be plus 15 B, our insides are negative seven times six B that's going to be minus 42 B and now to our last which we know are correct, we have negative seven times a positive five and that's going to be minus 35. So now let's see if the outsides and the insides add up to give us our negative 27 B. So bringing this down, we've got 18 B squared and then adding our outsides and insides, we have 15 B minus 42 B and that is negative 27 B and then bringing down our minus 35 this does match. So we have factored this correctly. It is parentheses three B minus seven, end parentheses, times new parentheses. Six B plus five. We look at our answer choices and that 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. 16y² − 6y − 27

Concetti chiave

Factoring Trinomials

Prime Trinomials

FOIL Method

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