Factor each trinomial, if possible. See Examples 3 and 4. 5a^2-7a...

Question

Factor each trinomial, if possible. See Examples 3 and 4. 5a^2-7ab-6b^2

Accepted Answer

Welcome back. I am so glad you're here. We are given the polynomial two X squared plus X Y minus 21 Y squared and we are asked to factor it. So the first step in our factoring journey is that we are going to look at all three of these terms here and ask if they have any factors in common that we can factor out in front. Well, only the first to have x variables and only the last two have y variables. So we can't factor out any variables. And then for our coefficients, we've got two unwritten one and a negative 21. Well, with that one, there, there are no whole numbers that we could factor out in front. So we can't factor out anything in France. We go to the next step of our factoring journey and create these two sets of parentheses. Now we're going to do the reverse of the foil process if possible. So we look at our first term here and we ask ourselves what times what gives us two X square, that's going to be our possible firsts. And for two X squared, while we know we're going to have an X times and X to get that X squared. And then we're also going to have the only factors for two or two and one. So we'll have a two and a one, but I'm not going to write one X there. I'll just keep this as an X. So for our first, we've got a two and a one. Now our lasts, we ask ourselves what times what gives us a negative Y squared. And for that similar to our X squared, we're going to have a Y times A Y for that Y squared. But we've got lots of options for that negative for our last order does matter. And to get a negative 21 we have many options. We have a negative three times a positive seven or a positive three times a negative seven. And because order matters, we could also have a negative seven times a positive three or a positive seven times a negative three. We also have the options of negative 21 times a positive one or positive 21 times a negative one or there's more, we could have a negative one times a positive 21 or a positive one times a negative 21. So how do we know which are the right factors to choose for our lasts? Well, it all comes down to when we go through the foil process and we add our outsides, we multiply our outsides and then add that to our insides. When we multiply those together, they have to add up to this plus xy, this one that's unwritten there, they have to add up to a positive one. So how do we know which ones are the right ones? Well, we can guess and check, we can try to figure it out. So, what we're going to do here is we're going to take our firsts and we're going to plug those in for our outsides and our insides and then we'll be multiplying those by our lasts. So we can pick our first pair from the last, we can take the two times a negative three and one times seven and then we'll be adding those products together to see if they equal a positive one. So let's test this out. We've got two times negative 32 times negative three is a negative six plus one times seven is a positive seven, negative six plus seven equals a positive one. We got it on the first try. Okay. We didn't have to test any of the rest of these. Sometimes you go through the entire list and sometimes you get lucky right away and sometimes you have an inkling knowing what it might possibly be. But okay now we've just got to plug these in into the right places. We've already got R two and R one plugged in. So now we need to get the negative three in the right place. The negative three is going on the outside. So the negative three needs to go over here in this position so that it will be multiplied as part of the outsides. Now, for our insides are inside last is this seven. So we're going to plug the plus seven over here and we can always foil this to check. So we take our first two X times X is two X squared. Our outsides to X times negative three, Y is minus six, Y are inside seven. Excuse me, not seven Y, it's seven okay. Two X times negative three, Y is negative six X Y there. Now the insides seven Y times X is plus seven X Y and the last seven Y times negative three Y is minus 21 Y squared. Now combining our like terms in the middle here, bringing down our two X squared negative six X Y plus seven X Y is plus one X Y or plus X Y and bring down our minus 21 Y squared. And yes, this matches what we had originally. So we factored it correctly. Now we have two X plus seven Y times X minus three Y, we look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

Factor each trinomial, if possible. See Examples 3 and 4. 5a^2-7ab-6b^2

Key Concepts

Factoring Trinomials

The Distributive Property

Zero Product Property

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