In Exercises 69–82, factor completely.

12x² + 10xy – 8y²

Question

In Exercises 69–82, factor completely.

12x² + 10xy – 8y²

Accepted Answer

Hello, today we're gonna be factoring the following trinomial. So what we are given is 60 X squared plus 69 X Y minus 21 Y squared. Now, before we factor this using the trial and error method, let's go ahead and check to see if we can factor out a greatest common factor. So if we take a look at the coefficients, we have 60, and 21, which all have a common multiple of three. So we can go ahead and factor out a three from the current trinomial and notice that not every term has an X in it and not every term has a Y in it. That means that the greatest common factor is only going to be three. So if we factor out a three from the current trinomial, what we are left with is 20 X squared plus 23 X Y minus seven Y squared. And now what we want to go ahead and do is factor this using the trial and error method. So what we're going to do is list out all the possible factors for 20 and all the possible factors for the last term. The factors for the first term are going to be called the P factors, which only consist of factors of positive numbers and the factors for the last term which I'm going to call the Q factors consist of factors of both negative and positive numbers. So what are the possible factors for 20 x squared? Well, we can go ahead and multiply X with 20 X, we can also try two X and 10 X And we can try five x and four x and for negative seven Y, well, seven is a prime number. But in order to get a negative number, we need to multiply a positive and a negative together. So there are two possible factors. We can have negative seven Y with positive Y or we can have positive seven Y with negative Y. Now, what I'm going to do is I'm going to draw two pairs of parentheses and within these pairs of parentheses, the first term is going to be the numbers from the P factors. And the last terms are going to be the numbers from the Q factors. And again, Q can be plus or minus. So let's go ahead and pick a pair of P factors and Q factors to test out and verify whether the factored form is tri is the original trinomial. So let's go ahead and try out five X and four X and let's go ahead and try negative seven Y with positive Y if we plug this in to our P Q method, what we end up with is five X minus seven Y times four X plus Y. And now, in order to verify whether this works or not, we're gonna go ahead and multiply the very first term with the very last term and the two middle terms together. And if there some gives us the middle term, which is going to be 23 XY, that's going to verify that this is the factored form of the trinomial. So five X times Y is going to give us five X Y and negative seven Y times four, X is going to give us negative 28 X Y and so five X Y minus 28 X Y is going to give us negative 23 X Y, but that is not the same thing as positive 23 X Y. So this is not going to be the correct factorization. Now, keep in mind we did get very close to positive 23 X Y. So let's go ahead and try the same combination, but let's switch the signs for both of the groups. So let's go ahead and try five X plus seven Y times four X minus Y. And so if we multiply the very first term with the last term, five X times negative Y is going to give us negative five X Y. And if we multiply the two middle terms together seven X times seven Y times four X is going to give us positive 28 X Y and 28 X Y minus five X Y is going to give us positive 23 X Y which matches up with the original middle term. So that means that five X plus seven Y times four X minus Y is going to be the factored version of the trinomial. But we want to be careful because keep in mind we factored out of three in the beginning. So now we want to go ahead and combine all of our factors together. Doing this is going to give us three times the quantity five X plus seven Y times the quantity four X minus Y. And this is going to be the completely factor version 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.

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In Exercises 69–82, factor completely. 12x² + 10xy – 8y²