Factor each polynomial. See Example 7. (5x-2)^3-8

Question

Accepted Answer

Hey everyone here we are asked to factor the following polynomial of the quantity of two U minus three cubed minus 27. 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. Now our next step is to just rewrite our original expression, making sure that both terms are written as cubes so it better matches our formula. So our original expression again we have the quantity of two U minus three cubed minus 27. And now rewriting, we just have to rewrite our second term. So we have the quantity of two U minus three cubed minus three cubed. Now our next step is to just compare our rewritten expression with our formula to find the values of X and Y. So starting with our value for X. What we see from the formula that our first term is going to be, our first cube term is going to be the value of X. So X in this case is just going to be two U minus three. And now our second term from the formula is the value of Why? So why is going to be three And now with the X and Y values clearly identified, we can go ahead and substitute them into the right hand side of our formula here to find the factor and form of our original expression. So again we have the expression of the quantity of two U minus three cubed minus 27. And now from our formula we see that this expression here is equivalent to the factored form where we have the quantity of X which again is to u minus three minus y. So we have a minus three and this is times the quantity of X squared which is the quantity of two U minus three squared plus X. Which is the quantity of two U minus three. This is times why which is just three. So times three and then we add y squared. So we just add three squared and now we just have to simplify this expression here. So our factored form we now have the quantity of two U minus six times the quantity of four U squared minus 12 U plus nine plus six U minus nine plus nine. Now we just have to make some other simplifications here. So we have the factored form of the quantity of two U minus six times the quantity of four U squared minus six U plus nine. And now our last step here is to check if we can factor anything else out of our expression so far. And in this case looking at our first set of parentheses, we see that we can factor out a two from both of these terms in the parentheses. So we can just factor out a two from this quantity all together. So we find that our expression of the quantity of two U minus three cubed minus 27 is the factored form of two times the quantity of U minus three times the quantity of four U squared minus six U plus nine. And with this we have found our fully factored form and we are left with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

Factor each polynomial. See Example 7. (5x-2)^3-8

Key Concepts

Polynomial Factoring

Difference of Cubes

Binomial Expansion

Watch next