Multiply or divide, as indicated. (xz - xw + 2yz - 2yw)/(z^2 - w^...

Question

Multiply or divide, as indicated. (xz - xw + 2yz - 2yw)/(z^2 - w^2) * (4z + 4w + xz + wx)/(16 - x^2)

Accepted Answer

Hey everyone here we are asked to simplify the expression by applying the operations indicated. So here our given expression we have to a c minus four A. D. Plus six B c minus 12 B. D. All divided by the quantity of four C squared minus 16 D squared. And this fraction is multiplied by another fraction. Where we have the numerator as six C. Plus to a C plus 12 D. Plus four A. D. And this is all divided by the quantity of nine minus a squared. So now our first step here in simplifying this expression is going to be two factor. And in our numerator is here we can factor by grouping. So starting with our first numerator of our first fraction, we see our first two terms we have to a c minus four A. D. Well in both of these terms here we have a common term of two A. So we can factor this out in doing this, we have to a times the quantity of c minus to D. And now moving on to our last two terms. In our first numerator here we have six B c minus 12 B. D. Well the common term here, we can factor out is just six B. So doing this, we have plus six B times the quantity of c minus two D. And now we can go ahead and move on to our second numerator here and looking at the first two terms again we have six C plus to a C. Well the common term here that we can factor out is just to see. So doing this, we have to see times the quantity of B. Or sorry three plus A. And now looking at our last two terms, we have 12 D. Plus four A. D. And we can factor out the common term of four D. So we have plus four D. Times the quantity of three plus A. And now moving on to the denominators for both of our fractions here we see that in both denominator we have the difference of two squares. So to factor both of the expressions we need to recall the difference of two squares formula which tells us that a squared minus B squared is equal to the quantity of a plus B. Times the quantity of A minus B. So now with this formula in mind we can factor both of our denominators. And starting with our first denominator of four C squared minus 16 D squared. According to our formula we can factor this expression as the quantity of two c minus four D. Times the quantity of two C plus four D. And now moving on to our second denominator where we have nine minus a squared. Well again according to our formula this expression will factor as the quantity of three minus a times the quantity of three plus A. So now we have finished our first round here of factoring but we can see in our numerator we can still group together more terms. So beginning with our first numerator we see that both terms now have the common expression here of c minus two D. So we can go ahead and group together to A and six B. So doing this, our first numerator becomes the quantity of two A plus six B. Times the quantity of C minus two two D. And now our denominator is just going to remain the same as the quantity of two C minus four D. Times the quantity of two C plus four D. And for our second fraction here again the numerator can be simplified further here by grouping together the two C. And four D. So doing this, we have the quantity of two C plus four D. Times the quantity of three plus A. And again our denominator is just going to remain the same as the quantity of three minus a times the quantity of three plus A. So now taking a look at what we have so far, we see that in our first numerator we can actually transform the expression further by factoring so doing this again, our first enumerators expression here, we have the quantity of two A plus six B times the quantity of c minus two D. Well, to further factor this expression, we see that in the first set of parentheses both terms have a two factor. So factoring out the two, we have two times the quantity of a plus three B times the quantity of C minus two D. And now with this factored out to, we can actually factor it into the second set of parentheses. And by doing this we see that this second set of parentheses, this c minus two D. Matches or will match the denominator of our first fraction here. So factoring in the to our numerator becomes the quantity of A plus three B times the quantity of two C minus four D. So now with this transformed numerator here we can go ahead and rewrite our expression. So doing this, our first fraction becomes the quantity of a plus three B times the quantity of two C minus four D. And this is all divided by the quantity of two C minus four D. Times the quantity of two C plus four D. And this fraction is multiplied by our second fraction here, which has a numerator of the quantity of two C plus four D. Times the quantity of three plus A. And this is all divided by the quantity of three minus a times the quantity of three plus a. So now since we have fully factored our original expression, we can go ahead and start canceling out all of the terms that can be canceled. So beginning with our first fraction here we see that we have a quantity of two C minus four D. Which also appears in our denominator. So we can cancel out both of these expressions. And now looking at our second fraction here, we see that the numerator and the denominator both have this quantity of three plus A. So they both cancel. And now comparing both of the fractions here since we are multiplying them, we see that in the numerator of our second fraction we have this to C plus four D. Which also appears in the denominator of our first fraction. So again both of these expressions cancel out. And so now we just need to rewrite our expression with the remaining terms that we have. So doing this, our expression becomes the quantity of A plus three B divided by the quantity of three minus A. And now we can further transform this expression that we have by making a always positive and we do this by multiplying by negative one. And so with this our final expression becomes the negative or minus times the quantity of A plus three B. All divided by a minus three. And this leaves us with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time

Multiply or divide, as indicated. (xz - xw + 2yz - 2yw)/(z^2 - w^2) * (4z + 4w + xz + wx)/(16 - x^2)

Key Concepts

Factoring Polynomials

Rational Expressions

Common Denominators

Watch next