Factor by any method. See Examples 1–7. (x+y)^3-(x-y)^3

Question

Accepted Answer

Hey everyone here we are asked to factor the following polynomial where we have the quantity of five M plus four N cubed minus the quantity of five M minus four N cubed. Now our first step here in factoring this expression is to start by simplifying this expression and we can do this by letting each term be equal to some variable. So we can start by letting the variable U be equal to our first term here. Which is this quantity of five M plus four N. And then for our other term we can choose another variable. I'm going to choose V. So we're going to let variable V be equal to the quantity of five M minus four M. And so now from here we just want to rewrite our original expression using these variables U. And V. So doing this, we see our original expression becomes u cubed minus v. Cute. And with this we can see our rewritten expression perfectly matches the identity. Where we have x cubed minus y cubed. Which is equal to the quantity of x minus Y times the quantity of X squared plus X, Y plus y squared. And so now we just want to rewrite our simplified expression according to this formula where we can see that X is going to be you and that Y is going to be V. So now rewriting our expression here, we had u cubed minus v cubed and according to our formula this is equal to the quantity of U minus V times the quantity of U squared plus U times v plus v squared. And so now our next step here is to just substitute back in our original values for U. And V into our factored expression here. So doing this we have the quantity of five M plus four N. For U minus the quantity of five M minus four N. And this is times the quantity of the quantity of five M plus four N squared plus the quantity of five M plus four N. Times the quantity of five M ma minus four N plus the quantity of five M minus four N squared. And so now our next step is to just start simplifying our factored expression here. And so next we have eight and the quantity of eight N times the quantity of 25 M squared plus 40 M n plus 16 and squared plus M squared minus 16 and squared plus 25 M squared minus M n plus 16 and squared. And so now we just need to complete our simplifications here on the right hand side set of parentheses. So doing this, we see our original Expression here turns out to factor into the quantity of eight n. times the quantity of or sorry, 75, 75 M squared plus and squared. And so with this we have fully factored out our original expression and therefore we are left with answer choice. B. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Factor by any method. See Examples 1–7. (x+y)^3-(x-y)^3

Key Concepts

Difference of Cubes

Binomial Expansion

Factoring Techniques

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