In Exercises 93–100, factor completely.−5x⁴y³ + 7x³y⁴ − 2x²y⁵

Question

Accepted Answer

Hello, today we're gonna be factoring the following trinomial. So what we are given is negative seven M to the fourth and cubed plus 10 M cubed and to the fourth minus three M squared and to the fifth. Now, before we factor this using the trial and error method, let's go ahead and see if we can factor out a greatest common factor. Now, when looking at the coefficients of the, of the terms, we have 7, 10 and three, none of these have a common multiple in each other. But keep in mind that the first term is negative, after we factor out a greatest common factor, we want the first term to be positive. So we can go ahead and factor out a negative one. Furthermore notice that there's at least two instances of M in each term. So we can go ahead and factor out an M squared from the trinomial and there's at least three instances of N in each term. So we can go ahead and factor out an N cubed from the trinomial. So if we go through and factor out negative M squared and cubed from the trinomial, what we are left with is negative M squared and cubed times the quantity seven M squared minus 10 M N plus three N squared. And now we're gonna go ahead and factor this using the trial and error method. So the trial and error method states that we want to go ahead and list out the first, the factors of the first term and the factors of the last term, the factors of the first term which are going to be the P factors are going to be all the possible factors of only positive numbers. So what are the factors that can multiply together to give us seven M squared? Well, since seven MS, since seven is a prime number, we can, the only possible factors are going to be seven M and M. Now what about the last term? The factors of the last term are going to be the Q factors and this can be any combination of factors of positive and negative numbers. So with three N squared three is a prime number which means that the possible factors are going to be three N times N or negative three N times negative N. Now, what I'm going to do is create two sets of parentheses. The first term of each set is going to be a number from the P factors. And the second term of each set is going to be a number from the Q factors which can either be positive or negative. So let's go ahead and pick a pair of factors to test. In order to verify the factorization, we're going to be using seven M and M for the P factors. And let's go ahead and test out negative three N and negative N for the Q factors plugging this in is going to give us seven M minus three N times M minus N. And now what we want to go ahead and do is multiply the very first and the very last number together and the two middle numbers together and add up their products. And if the sum of their products matches the middle term, which in this case is going to be negative 10 and MN, this is going to be the factorized form of the trinomial. So seven M times negative N is going to give us negative seven M N negative three N times M is going to give us negative three M N and negative seven N N minus three M N is going to give us negative 10 M N which matches the middle term of the trinomial. So seven M minus three N times M minus N is going to be the factor form of the trinomial. But we need to be careful because keep in mind we factored out a negative M squared times N cubed in the beginning of the problem. So putting all of our factors together is going to give us negative M squared and cubed times the quantity seven M minus three N times M minus N. And this is going to be the factored form of the original trinomial. And with that being said, the answer to this problem is going to be D. 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 93–100, factor completely.−5x⁴y³ + 7x³y⁴ − 2x²y⁵

Key Concepts

Factoring Polynomials

Greatest Common Factor (GCF)

Polynomial Degree and Terms

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