Factor by any method. See Examples 1–7. 64+(3x+2)^3

Question

Accepted Answer

Hey everyone here we are asked to factor the following polynomial of 512 plus the quantity of four B plus nine cubed. Now our first step here before we begin factoring this expression we want to simplify it. So doing this we can let the variable U be equal to our second term which is this expression of four B plus nine. And now doing this we just want to rewrite our original expression with the variable U. And so we get 512 plus U. Cubed. And now looking at our rewritten expression, we see that it's simpler to work with and we see that we can also apply the identity where we have X cubed plus Y cubed which is equal to the quantity of X plus Y times the quantity of X squared minus X. Y plus y squared. And so now we just want to rewrite our simplified expression from earlier to better match our left hand side of the formula. So originally we had the expression plus you Q. Now rewriting our expression. So it matches the formula. We have to change our first term here. So it is a cubed term. So doing this, we have a cubed plus you cute. And now we can see that our rewritten expression perfectly matches our formula here. So now we can compare the expression with the formula to find our values for X and Y. So starting with our value for X. Well we can see that X cubed is our first term here. So X cubed is going to be equal to a cubed from our expression which tells us that just X is going to be eight and now for the value for Y. Well from the expression we see that Y cubed is going to be the second term which in our expression is U cubed in. This tells us that why alone is just going to be you. And so now that we have identified the values for X and Y we can use them and substitute them into the right hand side of our formula. So we see that our expression from earlier which was plus you cute. We see from this formula that this expression is equivalent to the quantity of eight plus you times the quantity of eight squared minus eight. You plus U. Squared. And so now our next step is to just substitute back in our original declared value for you which as we recall is four B plus nine. So doing this, our factored expression now becomes the quantity of eight plus four B plus nine times the quantity of eight squared minus eight times the quantity of four B plus nine plus the quantity of four B plus nine squared. And so now our next step is to just start simplifying our factored expression here. And so next we get the quantity of 17 plus four B times the quantity of 65 four plus or sorry minus 64 minus 32 B minus 72 plus 16 B squared plus 72 B plus 81. And so now we just have to finish up our simplifications here by adding together like terms so doing this, we see our final factored expression becomes the quantity of 17 plus four B times the quantity of 73 plus 16 B squared plus 40 B. And with this looking at our expression here, we see that there aren't any other terms of which we can factor out. So this is our final factored expression of our original expression, which leaves us 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. 64+(3x+2)^3

Key Concepts

Factoring Polynomials

Cubic Expressions

Binomial Expansion

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