Exercises 143–145 will help you prepare for the material covered ...

Question

Exercises 143–145 will help you prepare for the material covered in the next section. In each exercise, factor completely.9b²x + 9b²y − 16x − 16y

Accepted Answer

Welcome back. I am so glad you're here. We are asked to perform complete factorization of the following polynomial expression. The expression we're given is 49 A squared, P plus 49 A squared Q minus P minus 64 Q. And our answer choices are answer choice. A open parentheses, seven A minus eight closed parentheses multiplied by open parentheses. Seven A plus eight closed parentheses multiplied by open parentheses. P minus Q closed parentheses. Answer choice B is open parentheses. Seven A minus eight closed parentheses multiplied by open parentheses. Seven A plus eight closed parentheses multiplied by open parentheses. Q minus P closed parentheses. Answer choice C open parentheses. Seven A minus eight closed parentheses multiplied by open parentheses. Seven A plus eight closed parentheses multiplied by open parentheses. P plus Q closed parentheses and answer choice D is open parentheses. Eight A minus seven closed parentheses multiplied by open parentheses. Eight A plus seven closed parentheses multiplied by open parentheses. P plus Q closed parentheses. The first step in factoring is that we are going to look at all of our terms and see if they have any common factors that we can factor out in front. So let's look at our coefficients. First, the first term coefficient is 49. The second term coefficient is also 49. The third term coefficient is 64 as well as the fourth term coefficient. Well, all four of them don't have anything in common, but we do know that the first two terms have something in common and the 3rd and 4th terms have a coefficient in common as well. Let's check the variables. So the first term has A squared and P, the second term has A squared and Q, the third term only has A P and the fourth term only has a Q. So there isn't a variable either that we can factor out of all of them. However, the 1st and 2nd terms both had an A squared. So we notice here we have four terms, we can factor by grouping. So that means we just look at the first two terms and ask ourselves what are the common factors? And we established that those are 49 A squared, those occur in both of the first two terms. So we put down 49 A squared and then it open parentheses and ask ourselves what's left of those first two terms when we factor out 49 A squared. So the first term, 49 A squared P divided by 49 A squared leaves us with just A P in the second term. 49 A squared. Q divided by 49 A squared that leaves us with just a Q. So we put plus Q in our parentheses and then close that parentheses. And now we factor by grouping in the second set here, which is the 3rd and 4th terms. So between negative 64 P and negative 64 Q, the greatest common factor is this negative 64. So we put that after our first set of parentheses And then right after negative 64 we put another open set of parentheses. And we do our factoring from the 3rd and 4th terms in the original polynomial, taking negative 64 P divided by negative 64 leaves us with just a positive P and negative 64 Q divided by negative leaves us with just plus Q in these parentheses. Now we can close the parentheses and we notice that both 49 A squared and negative 64 have a common factor this P plus Q. So we can factor out A P plus Q. We put open parentheses, P plus Q closed parentheses. And now that is multiplied by open parentheses. The first term 49 A squared that we factored out. And the second term that we factored out, which is minus 64 closed parentheses. We now have two factors here and we look at the second factor and we notice that we have a difference of squares. 49 is seven squared. A squared is A squared and it's a difference of so they're subtracting the second term here, 64 is eight squared. We recall from previous lessons that when we have a difference of squares, which could be written as open parentheses. A squared minus B squared, closed parentheses that we can factor that as open parentheses. A plus B closed parentheses multiplied by open parentheses. A minus B closed parentheses. And we just need to identify our A and B terms, fill those into the factored version of the difference of squares. And then we will have fully factored that difference of squares. So identifying our A and B terms, the A term is the first one getting squared here. That's the seven and the A. So A is equal to seven A for the second one here. Our B term, that's the one being subtracted and squared. Our B term is going to be equal to eight. So we have our A and B terms. Now let's fill those in, I'm going to bring down that first factor the P plus Q. So we have open parentheses, P plus Q closed parentheses multiplied by open parentheses. And now this is going to be our factored difference of squares filling in our A and B terms. So instead of A, it's seven A and then plus, and instead of B, it's eight closed parentheses multiplied by open parentheses instead of A, again, it's seven A and now minus and instead of B it is again, eight closed parentheses. And this is our polynomial fully factored. We look at our answer choices four and they'll be in a slightly different order. Open parentheses, P plus Q closed parentheses multiplied by open parentheses. Seven A plus eight, closed parentheses multiplied by open parentheses. Seven A minus eight closed parentheses. And that matches with answer choice. C well done. We'll catch you on the next one.

Exercises 143–145 will help you prepare for the material covered in the next section. In each exercise, factor completely.9b²x + 9b²y − 16x − 16y

Key Concepts

Factoring

Common Factors

Grouping

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