Factor each trinomial, if possible. See Examples 3 and 4. 36x^3+1...

Question

Factor each trinomial, if possible. See Examples 3 and 4. 36x^3+18x^2-4x

Accepted Answer

Welcome back. I am so glad you're here. We are given the polynomial eight X cubed plus two X squared minus 21 X and we're asked to factor it. Well, the first step in factoring is that we're going to look at these three terms and see if they have any common factors that we can factor out. Looking specifically at these coefficients 1st +82 and negative 21. They don't have any common factors. So now looking at our variables X cubed X squared and X, well, yeah, those are all divisible by an X. We can pull an X out and then dividing each of these by X eight X cubed divided by X leaves us with eight X squared two, X, two X squared divided by X leaves us with plus two X and negative 21 X divided by X leaves us with minus 21. And now that we factored out everything that we can in front, we can bring down this X and set up two sets of parentheses to see if we can factor this further using the reverse of the foil process. So first we look at this first term eight X squared, we ask ourselves what times what gives us eight X squared, what those will be will be our firsts. The X squared is going to be an X times and X. But with that eight, we've got options for the eight, the eight could be two times four or it could be eight times one because order doesn't matter for our first so we can leave it with those two options. And once we figured out our options for our first, then we go back up here and look at our third term, this negative 21 ask ourselves what times what will give us negative 21. Those will be our lasts For -21 to get a negative number. When we multiply, we know we'll have a positive times a negative in order does matter for the last. So we'll have quite a few options here. So a negative times a positive that could be a negative three times a positive seven or it could be a positive three times a negative seven. It could be a negative seven times a positive three or a positive seven times a negative three. But wait, there's more, we could have a negative 21 times a positive one or a positive times a negative one or a negative one times a positive or a positive one times a negative 21. So how do we know which one to choose? Well, this is where that going back to that foil process. When we do our outsides multiply those together and add that to our insides multiplied together, then that needs to equal this middle number up here, this positive two X or really the number we're focusing on here is it needs to be a plus two. So how do we figure out what combination gives us a positive to? Well, we can just start plugging things in and seeing what we get. I'm going to do a little bit of a shorthand version of that. You could go ahead and set up lots of parentheses and test things out. I'm going to do it underneath this word foil. So first of all, we're going to look at what our firsts could be for our outside. And I'm going to start with this first set, this first pair here, two and four for our first. So one first will be for the outside and one will be for the inside and then we can choose a pair for our last. I'll just start at the top negative three and seven. And so we will take the two from our firsts and multiply it by, we'll use the negative three from our lasts and that's going to get added to four from our firsts times the seven from our lasts. And now let's see what this adds up to two times negative three is going to be a negative six plus four times seven is 28. Well, 28 minus six. We've got a 22 that's not even close to positive two. So Now we know that pair is not going to work together and we can keep going down the lasts, keep trying for 24 for our first. So now let's go with positive three and negative seven. So I'll switch the signs here for a positive three and there's a negative seven. Well, two times three for our outsides. Well, that's going to be a positive six now and four times negative seven is going to be a negative 28. Well, now six plus negative 28 gives us a negative 22 that just switched the sign and it didn't get us any closer to a positive two. So this second pair here for our last does not work. Alright, let's keep trying going down our list for our lasts. So now let's work on negative seven for our outside and a positive three for our inside. So we've got two times negative seven will give us a negative plus four times three is a positive 12, negative 14 plus 12 gives us a negative to, hey, that's very close. And remember what we just learned from the other one when we switch the signs, we're going to switch the sign. So let's do that. Let's make our seven positive and our three negative. So two times positive seven is going to be a positive 14 and four times negative three will be a negative 12 and 14 minus 12. Gives us the positive to that we want. So this is our pair that we want, we want 24 and seven, negative three. Now let's get them in the proper positions in our parentheses. So for our outsides, that's here and here we've got two for the red spot for the first. So two X here in the first set of parentheses and then seven will be the last and that's going in the second set of parentheses is going to be a plus seven over here. All right now working on our insides, that's going to be here and here. So for insides, for a red spot for the first in this second set of parentheses, that's going to be four, so four in front of the X and then for the lasts, that'll be in the last section of the first set of parentheses. That'll be over here. That'll be a negative three. So we've got this set up. Now, what we can always do is we can always go back and check and make sure we've done this correctly so we can bring down this X and now we can multiply our first two X times four X gives us eight X squared and our outsides two X plus seven gives us plus 14 X, our insides negative three times four gives us a minus 12 X and our lasts negative three times seven is minus 21. We can combine our like terms in the middle. Bring down our X times parentheses. Bring down that A X squared 14 X minus 12 X gives us plus two X, bring down that minus 21 and yes, this does match what we started with. So we factored it correctly. Now we're looking for our answer choice that matches X times two, X minus three times four, X plus seven. That is answer choice. B well done. We'll catch you on the next one.

Factor each trinomial, if possible. See Examples 3 and 4. 36x^3+18x^2-4x

Key Concepts

Factoring Trinomials

Greatest Common Factor (GCF)

Polynomial Degree

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