In Exercises 31–38, factor completely.

4y³ + 12y² − 72y

Question

In Exercises 31–38, factor completely.

4y³ + 12y² − 72y

Accepted Answer

Welcome back. I am so glad you're here. We are asked to perform factorization in the given expression. We are given three R cubed minus 18 R squared minus R. Our answer choices are answer choice A three R multiplied by open parentheses. R plus one closed parentheses multiplied by open parentheses. R plus seven closed parentheses. Answer choice B three R multiplied by open parentheses. R minus one closed parentheses multiplied by open parentheses, R minus seven closed parentheses. Answer choice C three, R multiplied by open parentheses. R minus one closed parentheses multiplied by R plus seven closed parentheses and answer choice D three, R multiplied by open parentheses. R plus one closed parentheses multiplied by open parentheses. R minus seven closed parentheses. All right. The first step in factoring is that we look at all of our terms And we ask ourselves if they have any common factors and between three, negative 18 and negative 21, all of those coefficients, the greatest common factor between them is a positive three. Or in other words, we can factor out a three. Now looking at the variables we have R cubed R squared and R which is the same as R to the first. And all three of them have an R variable. And when we are looking for variables to factor out, we want to look for the one with the lowest degree, between 32 and one, the lowest degree is one. So we're dividing all three of them by or factoring out and R to the first, which is the same as R. Now, we know what, we're factoring out the three R. We want to figure out what's left behind once we factor that out. So we take the three R multiplied by an open parentheses and inside the parentheses is going to be what's left of each of those three terms, once we divide all three of them by three R. So let's take the first term, three R cubed divided by three R. Well, the three divided by three is one. So we can put a one as the first part of the first term inside of our parentheses. And for the R cubed divided by R, we recall from previous lessons that when we have the same base in this case, the R and we are dividing our variables, we are going to subtract their exponents. So in this case, it's going to be three minus one, which is two. So that's our new exponent. We have R squared or up here, we have our one time or excuse me, one multiplied by R squared which is R squared. We don't need to put that coefficient of one. We can just put R squared. Now, that's what's left of our first term. Now, we can take our second term negative 18 R squared divided by three, R, negative 18, divided by three is negative six. So we can put in our parentheses minus six and R squared divided by R to the first. That's two minus one which leaves us with one for our new exponent R to the first, which is the same as just R. So up here, it's negative six R inside of our parentheses. Now our third term from our trinomial, we can take negative 21 R divided by three R and that is going to be negative 21 divided by three is negative seven. So we can put minus seven in our parentheses. And then R to the first divided by R to the first, that's one minus one which is zero, our new X one is zero R to the zero power. Well, anything to the zero power is equal to one. So we have this negative seven multiplied by one. That's just negative seven. We don't need to put the multiplied by one. We can close our parentheses and we have that partially factored. However, looking at what's in our parentheses, we could factor that one step further. We can bring down this three R factor in front of our parentheses and then set up two open sets of parentheses. To further factor our R squared minus six, R minus seven. And in order to do that, we essentially do the reverse of the foil process. We start off asking ourselves what multiplied by what equals R squared. Those will be our first and it will be R multiplied by R that equals R squared. Now that we've got our first figured out, we ask ourselves what multiplied by what equals a negative seven and adds up to a negative six. And those are going to be a positive one And a negative seven. So we put in our parentheses are plus one with a one for our last there. And then the other set of parentheses are -7 with a negative seven for our last in the second set of parentheses. Well, now this is Foley factor. So we look at our answer choices for three, R multiplied by open parentheses, R plus one, closed parentheses, multiplied by open parentheses. R minus seven, closed parentheses. And that matches with answer choice. D well done. We'll catch you on the next one.

In Exercises 31–38, factor completely. 4y³ + 12y² − 72y

Key Concepts

Factoring Polynomials

Greatest Common Factor (GCF)

Quadratic Factoring

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