Factor each polynomial completely. See Example 6.4x² - 28x + 40

Question

Accepted Answer

Welcome back. I am so glad you're here. We're asked to rewrite the following expression by factoring it completely. Our expression is three N squared minus 33 N plus 84. Our answer choices are answer choice A three multiplied by the quantity of N minus four multiplied by the quantity of N minus seven. Answer choice B three multiplied by the quantity of N plus four multiplied by the quantity of N plus seven. Answer choice C three multiplied by the quantity of N minus multiplied by the quantity of N minus seven. And answer choice D three multiplied by the quantity of N plus 12 multiplied by the quantity of N plus seven. All right, taking a look at our expression. The first thing we want to do is see if there's anything that we can factor out of each of our terms. We have three N squared A negative 33 N and A positive 84. Well, let's just take a look at the coefficients first. We have three negative 33 and then the constant term positive 84 do three negative 33 and share any common factors. Well, three's only factors really are three and one but negative 33. Well, that's three multiplied by 11 and one of those would be negative. And then 84 we can pull a three out of 84 as well. That would be three multiplied by 28. So we can factor a three out of everything. Do all three of our terms share a common variable. They do not, 84 does not have an N attached to it. So we can't factor out any ends, but we can factor out that three. We put that in front of a set of parentheses. And now we divide each of our three terms by 33 N squared, divided by three leaves us with N squared, negative 33 N divided by three leaves us with a negative N and 84 divided by three leaves us with plus 28. Now we want to take a look at what's left inside of the parentheses and see if we can factor it any further. And what we can do here is the reverse of the foil process. So just keeping in mind that that three exists in front, but focusing in on what's in the parentheses for now. So to do the opposite of the foil process, we set up two open sets of parentheses, we can bring down that three. We begin by asking ourselves what multiplied by what gives us N squared. That's going to be our firsts. And for N squared, that's just going to be N multiplied by N next, we ask ourselves what multiplied by what gives us a positive 28. And because the coefficient of N squared was one, we can at the same time ask ourselves about our outsides. When we multiply those and add them to our insides, they need to add up to this negative 11 N. So the question we can ask ourselves is what two terms when multiplied, give us a positive 28 and when added, give us a negative 11 and that's going to be a negative four and a negative seven. Now, if you need to pause the video and foil this out to double check, you're welcome to do that. Otherwise that is our expression fully factored. So we look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Factor each polynomial completely. See Example 6.4x² - 28x + 40

Key Concepts

Factoring Polynomials

Quadratic Polynomials

Greatest Common Factor (GCF)

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