Factor each trinomial, if possible. See Examples 3 and 4. 14m^2+1...

Question

Factor each trinomial, if possible. See Examples 3 and 4. 14m^2+11mr-15r^2

Accepted Answer

Welcome back. I am so glad you're here. We are given this polynomial 14 P squared minus 19 P Q minus three Q squared. And we are asked to factor this will be recall from previous lessons. We are first going to look at these three terms here and see if they have anything in common. Any factors that we could factor out first, we'll take a look at these whole numbers 14, 19 and three. While three doesn't go into the other two, they don't share any common factors. So we can't factor out a whole number this first term and second term both have API but the third term doesn't. And the second and the third term both have a Q, but the first term doesn't. So there's nothing that we can factor out in front for this one. So that's step one of factoring. Next. We are going to set up two sets of parentheses And we are going to do the reverse of the foil process. We're going to factor these. And so first, we look at this first term here and we say what times what equals 14 p squared. And we know that we're going to have a P times A P to get that P squared. But what about 14? Well, we've got options for 14, 14 could be seven times two or it could be 14 times one order doesn't matter for the first one. So we can leave it with those two options. Next. Once we've gotten those possible factors figured out for the first one, we take a look at the last term here, this negative three Q and we're going to say what times what gives us our negative three Q, we're looking for our lasts. And similarly to that P squared, we know we're going to have a queue times a cue for our Q squared. But what about that? Negative three? Well, for a negative number something times something to equal negative number one will be positive and one will be negative and three doesn't have a whole lot of factors. We know that it's got a three and a one. So we could try negative three times a positive one or we could try a positive three times a negative one order does matter for the last. So we also need to try a negative one times a positive three and a positive one times a negative three. And so those are all of our different options. But why does it matter? Why do we have to go through all of these options? Well, when we take our outsides multiply those together and add those to our insides, then we need that to add up to a negative P Q when we go through our foiling process. So how do we figure out which ones will add up to be a negative 19? Well, we get to just try different ones and find out what we can do is we can take from our first. So the red ones and so we can take a seven and two for our red ones. And then we can go and choose the first pair from our last. We can try negative three and one. And so we can take Seven times and then the first one of our lasts -3. And then we'll add that to two from the first times are positive one from our lasts and then we'll see if that adds up to be a negative 19. So seven times negative three is negative 21 plus two times one is two, negative 21 plus two is a negative 19. And sometimes you will be extraordinarily lucky like that and you will pick the right one on the first time or maybe you'll have had an inkling that it might have to be something like that. So this time we got it straight away. Now we just need to put everything into the correct places. So for the outsides, we know that our red first is the seven. So that's going to go in front of this p in the first set. Of parentheses. So we have a seven P there. And then again, for our outside, the last outside is negative three. Recall that the outside is over here. It doesn't go in the same set of parentheses. So this is minus three Q and then those who will get multiplied together. Now, for the insides, that's these two for the first, that's going in the red position, that's going to be too. So that goes in front of the P in the second set of parentheses. And then the one goes in front of the queue for the first set of parentheses for the last position. And we would have a plus one. I'm not going to put the one there. I'm just going to make the Q bigger because one Q is Q and then that will match with our answer choice. If any of this is confusing, you can always just plug numbers right into the parentheses and then do the foil process to check. So let's do that checking. So first seven P times two P, that's 14 P. Our outsides seven P times negative three Q is negative 21 P Q R insides Q times two P is plus two PQ and our last Q times negative three Q is negative three Q squared. Now combining our like terms in the middle here, we can bring down 07 P times two PS 14 P squared, excuse me. And we'll bring down that 14 P squared. And the negative 21 PQ plus two PQ is negative 19 P Q and bring down our negative three Q squared. And yes, this does match what we started with. So we factored it correctly. So we've got seven P plus Q times to P minus three Q. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

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Factor each trinomial, if possible. See Examples 3 and 4. 14m^2+11mr-15r^2