Factor by any method. See Examples 1–7. 4z^4-7z^2-15

Question

Accepted Answer

Hey everyone here we are asked to factor the following polynomial of seven. Z to the fourth power minus 31 Z squared minus 20. Now, taking a look at our given expression here, we see that it would be rather complicated to start factoring right away since we're used to factoring expressions where the highest power is two rather than four like it is here. So our first step is to simplify our given expression and we can do this by letting the variable U be equal to Z squared. So with this we now need to rewrite our original expression using the variable U. So doing this, we have seven times U squared minus 31 times U minus 20. So now looking at our rewritten expression, we see that it would be much easier to simplify this X. Sorry, factor this expression. So we need to consider the factors as the quantity of A. U plus B times the quantity of C. U plus D. And now our next step is to find the value of a. C and find the value of B. D. Well, starting with a C. We see that A C is going to be the coefficient of our first term which is seven. And now for our value from B. D. We'll be D is just the value of our third term here which is negative 20. And so now our next step is to find the factors of both of these terms here. So we can start with a C which again has the value of seven. And while the factors of seven are just one and seven or negative one and negative seven. And then moving on to the factors of B. D. Which is negative 20. We have 20 and negative one. We have negative 20 and positive one. We have 10 and negative two as well as negative and positive two. We also have four and negative five and negative four and positive five. So now that we have all of the factors identified, we have to go through a trial and error method until we find a set of factors that return our expression from earlier and after this trial and error process here we find that the factored form of our sin simplified expression here turns out to be the quantity of seven U plus four times the quantity of U minus five. So now our last step here is to just substitute back in the value for you. So doing this, we see that our factored form of our original expression turns out to be the quantity of seven Z squared plus four times the quantity of Z squared minus five. And with this we have found our fully factored form of our original expression here which is answer choice D. 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. 4z^4-7z^2-15

Key Concepts

Factoring Polynomials

Quadratic Form

Zero Product Property

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