Factor each trinomial, if possible. See Examples 3 and 4.

Question

Factor each trinomial, if possible. See Examples 3 and 4. $12 s^{2} + 11 s t - 5 t^{2}$

Accepted Answer

Welcome back. I am so glad you're here. We are given the polynomial four S squared minus 23 S T plus T squared and we are asked to factor it. Well, the first step in factoring is we look at all of our terms here and we ask ourselves if they share any factors that we can factor out in front. Well, only the first two terms have S variables and not the third one and only the last two terms have T variables and not the first one. So we can't factor out any variables. How about 4, 23 15? For these whole numbers, they don't share any common factors. So there's nothing that we can factor out in front. So the next step in our factoring journey is that we're going to set up these two sets of parentheses and we're going to do the reverse of the foil process. So we're going to ask ourselves what times what gives us this four S squared. Those will be our firsts. So for four S squared, we have some options. We know that to get that S squared, we're going to have an S times and s, but for the four, it could be a couple of different things. We could have four times one or we could have two times two order doesn't matter for the first. So we don't have to include one and four as possibilities. So we've got options for our firsts. Now, let's take a look at our last, we say what times what equals a positive T squared. And again, we've got options similar to the S squared for R T squared. We're going to have a T times A T. But what will give us that positive? 15? Well, to get to multiply to have a positive number, we could have a positive times a positive or a negative times a negative. And we're going to look at our middle term here for a clue because this is a negative 23. We're going to need to have a negative times a negative to get our positive 15 T squared. So we'll look at negative factors of 15 To get 15. We might have a negative five times a negative three. But order does matter for our last. So we might also have a negative three times a negative five or we could have a negative 15 times a negative one or a negative one times a negative 15. Those are all possibilities for our lasts. Now, The next step here is we need to figure out what combination of these factors to plug in to our parentheses so that they will add up to be negative 23st when we add our outsides plus our insides in the foil process. So how do we figure that out? Well, we just start plugging things in. So I'm going to be doing kind of a shorthand of that underneath here. What we do is we start, um, we can pick R four and R one for our firsts. So we'll have our four and our one for our first. And then we choose a pair for our lasts and we can choose negative five and negative three for the first pair that we're going to check. And then we're going to multiply our outsides together and multiply our insides together and then add what we get at our two products. So we have four times negative five, that's negative 20 plus one times negative three is negative three. So negative 20 plus negative three equals a negative 23. And that's exactly what we were looking for for our negative 23 S T. So these are the correct pairs and now we just have to plug them into the correct spaces in our parentheses. So for our outsides, looking at the outsides here here and here, we'll have a four for the first and a negative five for the last. And now looking at our insides, we will be plugging those in here and here. So for our first, we have a plus one, positive one. And for our last, we have a negative, oops, that's the wrong spot. Okay. For our first that's in red. Our first is a one so that the one S and our last is a negative three. So that goes here in the first set of parentheses. So we've got these plugged in. I'm going to substitute for this one. S that's the same as just plain S, so I'm going to plug that in. We can always check this by foiling it out. Um longhand. So four S times S is for S squared doing our firsts. Now our outsides for us times negative five T is minus 20 S T, our insides negative three T times S is minus three S T and now doing our last negative three T times negative five T is plus 15 T squared. Combining our like terms in the middle here bringing down round four S squared, negative 20 S T minus three S T. Is that minus 23 S T that we worked so hard for then bring down our plus 15 T squared and yes, this matches what we started with. So we know we factored it correctly. So our final answer is four S minus three T times S minus five T. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Factor each trinomial, if possible. See Examples 3 and 4. $12 s^{2} + 11 s t - 5 t^{2}$

Key Concepts

Factoring Trinomials

Multiplying and Splitting the Middle Term

Factoring by Grouping

Watch next

Factor each trinomial, if possible. See Examples 3 and 4. 12s2+11st−5t212s^2+11st-5t^2

Key Concepts

Factoring Trinomials

Multiplying and Splitting the Middle Term

Factoring by Grouping

Watch next

Factor each trinomial, if possible. See Examples 3 and 4. $12 s^{2} + 11 s t - 5 t^{2}$