Factor each trinomial, if possible. See Examples 3 and 4. 12x^2-x...

Question

Factor each trinomial, if possible. See Examples 3 and 4. 12x^2-xy-y^2

Accepted Answer

Welcome back. I am so glad you're here. We are given the polynomial five X squared minus 12 X Y minus nine Y squared and we are asked to factor it. So we recall from previous lessons that the first thing we'll do is take a look at these three terms here and see if they have anything in common, any factors in common that we can factor out. And in particular that they don't all have an X only the first to do and they don't all have a why only the second to do. So we can't factor out any variables and looking at 5, 12 and nine, those don't share any common factors either. So we're not going to factor anything out in front for this one. So the next step is we start with these two sets of parentheses and we're doing basically the reverse of the foil process. We ask ourselves what times what gives us five X squared. Those are our firsts that we're looking to assign and for five X squared. Well, we know we're going to have an X times an X to get that X squared. Now, for the five, we could have five times one or one times five. Order doesn't matter for the first. So I'm just going to choose five times one and plug those in five times 1 because those are our only options. The only factors for our firsts. All right, now that we've got our first figured out, now we look at our last, this negative nine. Why squared, what times what gives us negative nine? Why squared for our lasts? And we've got a lot of options for this one because order does matter for the last. We know we'll have a Why times a Why to get that Y squared. But for that negative nine, we've got options, we could have negative nine times one or we could have nine times a negative one because we'll have to have a positive times negative to get that negative nine. But order does matter. So we could also have a negative one times nine or a positive one times a negative nine. And then we've also got negative three times a positive three or positive, three times a negative three. So there are all of our possible options for our lasts. Now, why does it matter? Why are there so many different options for our lasts? Because they have to, when we have our outsides plus our insides, they are going to have to add up to negative 12 X Y. So how do we figure out which combinations add up to a negative 12 X Y while we can just start plugging them in and see what works. So for our outsides, we're always going to take um the first of our pair here for our first. So we'll take a five and we're going to multiply it by something we could plug in over here for our outsides. So something from the lasts and we'll just start with negative nine. So five times negative nine and that will be plus and then we take from the corresponding pair for our first one times the corresponding pair from our lasts one. So five times negative nine is negative 45 plus one times one is one. No, that's not the negative 12 that we're looking for. So we know this first pair is not going to work and now we can try the next pair. So we've got, now we've got from the first five times and now looking at this next pair for our last positive nine plus one from the first times negative one from the last, well, five times nine, that's 45 plus one times negative one is negative one. No, that's not negative 12. So that pair isn't going to work either. Now, we can continue so we can keep our firsts aren't going to change, so we can keep our five and our one and we'll just change what we're picking from our last. So we have five times. Now let's do five times negative one and then from the corresponding pair for our last one plus one times times negative one is negative five plus one times nine is nine, negative five plus nine is going to be a positive four. Nope, that is not going to add up two are negative 12. So we can cross off that ordered pair for our lasts. And now if we wanted to test that next 11 and negative nine, we know that it's going to be very similar. So five times one and plus one times negative +95 plus negative nine gives us a negative four. Really? All we're doing is switching the sign here. So we know that this pair isn't going to work either. Okay. Now the three's it's got to be some combination of the threes. So we can do again from our first for our outsides. We have five times and then we'll pull from the first part of our lasts five times negative three plus and then it'll be one from the first times and then three from the last. So five times negative three is negative 15 plus one times three is three, negative 15 plus three is negative 12. There it is okay. So we know what our first star we have those filled in and now we have to fill in our last into the correct spots and note here for our outsides, the one that has to be the outside is negative three. So our outside that's over here. So we have a minus three Y right there and then for our insides, that's going to be the one times positive three. So that'll be a plus three there. Okay. I am going to rewrite this, this one X that can just be an X because that will correspond better with the answer choices. So we've got five X plus three Y times X minus three Y. And if any of this was confusing, you can always do the foil process to check it. You do five X times X gives you five X squared for the first and then you do the outsides five X times negative three Y is minus 15 X Y and then the insides three Y times X is plus three X Y and then you do the lasts three Y times negative three Y is negative nine Y squared. And then you add up these outsides and insides and you get five X squared. Oops, I wanted to change colors there. I apologize. You get five X squared negative 15 X Y plus three X Y is minus 12 X Y minus nine Y squared. And then you know that that's what we started with. So looking at our answer choices, five X plus three Y times X minus three Y matches with answer choice. A well done. We'll catch you on the next one.

Factor each trinomial, if possible. See Examples 3 and 4. 12x^2-xy-y^2

Key Concepts

Factoring Trinomials

Difference of Squares

Grouping Method

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