Factor each polynomial. See Example 7. (3x+4)^3-1

Question

Accepted Answer

Hey everyone here we are asked to factor the following polynomial of the quantity of four a minus five cubed minus one. Now our first step here in factoring this expression is to apply 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. So now our next step is to just rewrite our original expression so that it better matches our formula. So originally we have the quantity of four a minus five cubed minus one. And now for rewriting we see our first term here is already written as a cube. So the only change we have to make is to write our second term as a cube. So again that it better matches our formula here. So our expression is now the quantity of four a minus five cubed minus one cubed. So with this with our rewritten expression we can go ahead and compare it to our formula to find the values for X and Y. So starting with our value for X. Well from the formula, we see that X is going to be our first cubed term here and so we see that X is going to be for a minus five and then we see again from our formula that Y is going to be our second term. So why is just going to be one? And so now with our X and Y values clearly identified, we can go ahead and plug them into the right hand side of our formula here. So from our expression of the quantity of four a minus five cubed minus one. We have the factored form which we know is the factored form from our formula here. So our original expression becomes the quantity of four a minus five minus one times the quantity of the quantity of four a minus five squared plus the quantity of four a minus five times one plus one squared. And so now we just have to start simplifying our factored expression here. So simplifying, we have the quantity of four a minus six times the quantity of 16 A squared minus A plus 25 plus four A minus five plus one. And now we just have some other simplifications here to make to our second set of parentheses. So doing this we have the expression of the quantity of four a minus five times the quantity of 16 a squared minus 36 A plus 21. And so now our last step here is to check our two simplified sets of parentheses here and see if there's anything else that we can factor out. Well seen in our first set of parentheses this five should actually be a six here. So we have four a -6. And well we can see that we can divide both terms by two, so we can go ahead and factor out a two from this first set of parentheses. So doing this we have our final factored form of the expression where we had the quantity of four a minus five cubed minus one and our factored form is now two times the quantity of two a minus three times the quantity of 16 A squared minus 36 A plus 21. And so now we have found our fully factored form of our original expression and 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 each polynomial. See Example 7. (3x+4)^3-1

Key Concepts

Polynomial Factoring

Difference of Cubes

Binomial Expansion

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