In Exercises 69–82, factor completely.

4x³y⁵ + 24x²y⁵ – 64xy⁵

Question

In Exercises 69–82, factor completely.

4x³y⁵ + 24x²y⁵ – 64xy⁵

Accepted Answer

Hello, today we're gonna be factoring the following trinomial completely. So what we are given is three X to the third, Y to the fifth plus 12 X squared Y to the fifth minus 135 X times Y to the fifth. Now, in order to factor this problem, let's go ahead and see if we can factor out a greatest common factor. So taking a look at the leading coefficients, we have 312 and 135 which all share a common multiple of three. So we can go ahead and factor out a three from the trinomial. Furthermore, each term here has at least one X in every term. So we can go ahead and factor out an X from the trinomial and every term has Y to the fifth. So we can go ahead and factor out Y to the fifth from the trinomial. So if we factor out three X Y to the fifth from the original trinomial, what we are left with is three X Y to the fifth times the quantity X squared plus four X minus 45. And now what we can go ahead and do is factor this using the trial and error method. So let's go ahead and list out the possible factors for X squared while these factors are going to be the P factors and the only possible factors for X squared is going to be X times X. And now what are the possible factors of negative 45? Well, those factors can either be one and negative 45 po negative one and 45 or we can use negative five and nine or five and negative nine. So I'm gonna go ahead and create two pairs of parentheses. And what's gonna go in these pairs of parentheses are the numbers from the P factors and the numbers from the Q factors and the Q factors can be positive or negative. So let's go ahead and pick two pairs of P and Q factors and test to see which one gives us the correct factorization. So we're going to try out X and X and let's go ahead and try negative five and nine. Putting this into the parentheses is going to give us X minus five times X plus nine. And in order to verify that this is the correct factorization, we can go ahead and multiply the first and last term together and the two middle two terms together. And if the sum of the products gives us the original middle term, which is positive for X, then this is going to be the correct factorization. So X times nine is going to give us nine X and negative five times X is going to give us negative five X nine, X minus five X is going to give us positive four X which matches the middle term. So X minus five times X plus nine is going to be the correct factorization for the trinomial. But keep in mind that we factor out three X Y to the fifth in the beginning. So combining all of our factors together is going to give us three X Y to the fifth times X minus five times X plus nine. And this is going to be the completely factored version of the original trinomial. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 69–82, factor completely. 4x³y⁵ + 24x²y⁵ – 64xy⁵

Key Concepts

Factoring Polynomials

Greatest Common Factor (GCF)

Polynomial Degree and Terms

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