In Exercises 16–17, factor completely.

24x³y + 16x²y − 30xy

Question

In Exercises 16–17, factor completely.

24x³y + 16x²y − 30xy

Accepted Answer

Hello, today we're gonna be completely factoring the given expression. So what we are given is 24 A cube times B minus 44 A square times B minus 140 A times B. Now, before we factor the given expression, let's go ahead and see if we can factor out a greatest common factor. Now, we're gonna take a look at the coefficients of the three terms. We have 24, 44 and 140. And all three of these terms share a common multiple of four. So we can go ahead and factor out A four from the given expression. Furthermore, there is at least one A in each of the given terms and there is at least one B in each of the given terms. So we can also factor out an A B from the given expression. So if we factor out four A B from the given expression, we can go ahead and rewrite it as four A B times the quantity six A squared minus A minus 35. And now what we're going to do is we're going to factor this trinomial using the trial and error method. Now, the trial and error method states that we want to go ahead and write the terms or the factors of the first term and the last term, the factors of the first term which I'm going to call the P factors are going to be any combination of factors of only positive numbers. So what are two factors that we can multiply together to give us six A squared while those factors can be six a times A or we can use three, a times two A. And now we need to list out the possible factors for -35, which in this case, I'm going to call the Q factors. Now, the possible factors for negative 35 can consist of positive and negative numbers and those are going to be one and negative 35 or negative one and positive 35. We can also use positive five and negative seven or negative five and positive seven. So, what I'm going to do now is I'm going to create two pairs of parentheses. And what is gonna go in these pairs of parentheses are going to be the leading terms P which are the numbers from the P factors. And the last terms Q which are going to be from the Q factors. And what we're going to do is we're going to pick a different combination of P and Q factors to plug in these spin into these parentheses and verify that these are the factored form of the given trinomial. So let's go ahead and try three A and two A and let's go ahead and try five and negative seven. So if we plug this into our pairs of parentheses, what we have is three, a plus five times 2 A -7. And what we're going to do now is we're going to multiply the first term with the very last term and the two middle two terms together. And if the sum of these products add up to give us negative 11 A, then that is going to confirm that this is the factored form of the trinomial. So three A times negative seven is going to give us negative 21 A and 5 times 2 a is going to give us positive 10 a. So negative 21 A plus 10 A is going to give us negative 11 A which is going to be the middle term in the trinomial. So this verifies that three A plus five times two A minus seven is the factored form of six A squared minus 11 A minus 35. But we need to be careful because keep in mind we factored out A four A B in the beginning of the problem. So now what we're going to do is we're going to combine all our factors together and doing so is going to give us four A B times the quantity three A plus five times the quantity two a minus seven. And this is going to be the completely factored form of the given expression. And with that being said, the answer to this problem is going to be a. 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 16–17, factor completely. 24x³y + 16x²y − 30xy

Key Concepts

Factoring Polynomials

Greatest Common Factor (GCF)

Combining Like Terms

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