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.5x² − 16x + 3

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 the answers correct by using the foil method. The tri no meal we're given is seven B squared minus 67 B plus 36. Our answer choices are answer choice. A parentheses, seven B minus nine and parentheses times new parentheses, B minus four and parentheses. Answer choice, B seven B minus four, end parentheses, times new parentheses, B minus nine and parentheses. Answer choice C parentheses seven B minus 12 and parentheses times new parentheses, B minus four and parentheses and answer choice. D the given try no meal is prime. Alright. So the first step, in fact, during a training meals, we take a look at each of these three terms and ask ourselves if there are any common factors that we can factor out in between seven, negative 67 positive 36. They don't share any common factors and they don't all have bees with them either. So we can't factor out any variables. So our next step is that we set up two sets of parentheses and we're essentially going to do the reverse of the foil process. So we're going to ask ourselves what times what equals seven B squared. And for that B squared, we know that's going to be A B times A B for that seven. That can only be one times seven or seven times one order doesn't matter for the first because it could end up working either way. But we look at our answer choices and they all have the seven B first. So we'll just keep it with seven B and then times one B which is just be so I'll center this B And now we've got our first taken care of. And next, we ask ourselves what times what equals a positive 36. Those are going to be our lasts. So our firsts were first and then now we're looking at our lasts. So positive 36, we've got a lot of options for factors that equal positive 36. We could have one times 36. We could also have negative one times negative 36. We could also have because order matters for the last, we could have 36 times one or negative, 36 times negative one. We could also have two times 18 or negative, two times negative 18 or 18 times two or negative, 18 times negative two. We could have three times 12 or negative three times negative 12 or 12 times three or negative, 12 times negative three. We could have four times nine, negative, four times negative 99 times four, negative, nine times negative four or we could have six times six or negative, six times negative six. There are so many options. How do we know what to choose where to even start? Well, what we're going to be doing is we're taking our firsts from the first set of parentheses times the last from the second set of parentheses. Those are our outsides and we're going to add those to the insides which are the last from the first set of parentheses times the first from the second set of parentheses. And they must add up to equal negative 67 B this middle term from our train o meal. So we can just start filling in all of these different last to see what combination adds up to negative 67 B. However, this is multiple choice and we can narrow it down a bit by looking at what they've chosen for these answer choices. So answer choice, A for the last they've chosen negative nine and negative for, let's just go ahead and try those. So we've got for our first set of parentheses, we have seven B minus nine. And for the second set of parentheses, we have B minus four. Now, let's foil this. We do our first which are seven B times B which is seven B squared by design. We knew that would work. Our outsides are seven B times negative four and that's going to be a negative 28 B. Our insides are negative nine times B which is negative nine B and our last are negative nine times negative four which is positive 36. All right. Now we combine the like terms in the middle here, negative 28 B minus nine B that's going to equal negative 37 B. Nope, that's not the negative 67 that we wanted. So we can eliminate these choices of - and -4. For our last, we can eliminate those. And now let's try what they've got for answer choice B you might notice that all of these are the negative options and that's because they have to add up to a negative 67. So really, we can eliminate all of the positive options for our lasts. And that would be a key to help you if you had to go through and had no multiple choice selections to narrow this down. But now let's look at the last from answer choice B which are negative four times negative nine. So very similar to what we just did. We've got seven B minus four times B minus nine. Now we know our first and our last will multiply correctly. So let's just focus on the outsides and the insides, the outsides seven B times negative nine is negative 63 B. This is looking good and then our insides negative four times B that's minus four B so we can bring down our first, we have seven B squared. Combine our middle like terms here, negative 63 B minus four B is negative 67 B and then bring down our last plus 36 this matches. So we have factored it correctly. We've got seven B minus four times B minus nine which matches answer choice B 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.5x² − 16x + 3

Key Concepts

Factoring Trinomials

FOIL Method

Prime Trinomials

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