In Exercises 17–38, factor each trinomial, or state that the trin...

Question

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 6x^2−7xy−5y^2

Accepted Answer

Welcome back. I'm so glad you're here. We have got a polynomial. We've got eight X squared minus six X y minus two Y squared. And we are to factor this. And if we can't to identify that it is prime, that's as factored as it's going to get. So the first step is we're going to look at each of these three terms here, eight X squared, negative six X, Y and negative two Y squared. And say, do they have anything in common that we can factor out in front? And yeah, they do. They've all got a two in them. So now we take a two factor that out front and say, okay, eight X squared divided by two. That leaves four X squared negative six X Y divided by two. That leaves us negative three X, y and negative two Y squared divided by two. Well that's minus one Y squared. You don't have to put that one there, but I'm going to leave it there for this next step so we can see it the next step. We've got we bring down this two and now we're going to factor it and factoring it isn't too bad when there's no number in front of our X squared in front of this first term here. But it's a little bit more of a process when we have a number in front of that. So the first question we're going to ask ourselves is what times what equals four X squared. And those are going to be our firsts if we go back later and foil this to check. So what times what gives us four X squared. But we have options actually that could be a four X times a one X. Or it could be a two X times a two X. So those are options here. And the next question we ask ourselves is what times what gives us a negative one Y squared. And those are going to be our lasts when we will go back and foil this to check. So what times what gives us negative one, Y squared. Well that could be a positive Y times a negative Y. Or it could be a negative Y times a positive Y. And so at this point, what you could do is you could start filling in these options and checking them just by foiling. So you could say four X plus y, times one x minus y. You can check it with foil and see if that equals our previous polynomial here in the parentheses. But we can also think about this before we start guessing checking and we can say, well we know that our Outsides plus our insides are going to add up to this negative three X. Y here in the middle. And since over here the wise, those are just ones ones positive and one's negative. We can think about, okay, those are going to get multiplied by our numbers here. And what would add up to a negative three Xy and well too, when it gets multiplied by one plus two. When it gets multiplied by one. If one's positive, one's negative. Those are not going to add up to be a negative three. So we're probably working with this four and the one. So now we just need to say okay, which one has to be bigger? Which one has to be negative and which one has to be positive And to get to negative three xy that four needs to be negative. So the thing that the four needs to be multiplied by or are outside has to be negative. So that's going to be this minus Y here. And the thing that the X. Has to be multiplied by in our insides. In order for it to be positive is going to be our why here are positive. Why? So we can put these in parentheses and now we can foil this to check and see if we did it correctly. It's always good to check in these cases. So our first four X times one X. That is four X squared. That matches our outsides. We've got four X times negative Y. Is going to be negative for X. Y. Our insides we've got a Y times A one X. Is plus one X. Y. And r lasts. We've got a Y. Times and negative Y is going to be negative Y squared we combine like terms in the middle, bring down our four X squared negative four X Y plus a one X Y is going to be that negative three X. Y. That we were looking for and that bring down that negative Y squared, which is the same as negative one Y squared. So yes, these are correct two times four X plus Y times one X. We can just write X minus Y. That is our final and correct answer. And we look at our answer choices and that matches with answer choice. A well done. We'll catch on the next one.

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 6x^2−7xy−5y^2

Key Concepts

Factoring Trinomials

Prime Trinomials

The Discriminant

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