Dividing polynomials by linear factors
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- [Instructor] We are told the polynomial p of x is equal to four x to the third plus 19 x squared plus 19 x minus six has a known factor of x plus two. Rewrite p of x as a product of linear factors. So pause this video and see if you can have a go at that. All right, now let's work through it together. So if they didn't give us this second piece of information that as a known factor of x plus two, this polynomial would not be so easy to factor. But because we know we have a known factor of x plus two, I could divide that into our expression and figure out what I have left over, and then see if I can factor from there. So let's do that. Let's divide x plus two into our polynomial. So it's four x to the third power plus 19 x squared plus 19 x minus six. And we've done this multiple times already. We look at the highest-degree terms. X goes into four x to the third. Four x squared times, I put that in the x squared or the second-degree column. Four x squared times x is four x to the third. Four x squared times two is eight x, eight x squared. And then I wanna subtract these from what I have up here. So I'll subtract. And then I'm going to be left with 19 x squared minus eight x squared is 11 x squared. And then I will bring down, bring down the 19 x. So plus 19 x. And so once again, look at x and 11 x squared. X goes into 11, x squared 11 x times. So that's plus 11 x. 11 x times x is 11 x squared. 11 x times two is 22 x. Need to subtract these from what we have in that teal color. And we are left with 19 minus 22 of something is negative three of that something. In this case, it's negative three x. And then we bring down that negative six. And then we look at, once again, at the x and the negative three x. X goes into negative three x negative three times. And so negative three times x is negative three x. Negative three times two is negative six. And then if we wanna subtract what we have in red from what we have in magenta. So I could just multiply them both by negative. And so everything just cancels out and we have no remainder. And so we can rewrite p of x now. We can rewrite p of x as being equal to x plus two times all of this business, four x squared plus 11 x minus three. Now we're not done yet because we haven't expressed it as a product of linear factors. This one over here is linear but four x squared plus 11 x minus three, that's still quadratic. So we have to factor that further. And let's see, there's a couple of ways we could approach it. We could use, well, we could try with this and the quadratic formula or we could factor by grouping. And to factor by grouping, and the whole reason we have to factor by grouping is we have a leading coefficient here that is not one. And so we need to think of two numbers whose product is equal to four times negative three. So we have to think of two numbers, let's just call them a and b. A times b needs to be equal to four times negative three, which is negative 12. And a plus b needs to be equal to 11. And so the best that I can think of, they have to be opposite signs 'cause their product is a negative. So If I had negative, if I had, if I had positive 12 and negative one, that works. If a is equal to negative one and b is equal to positive 12, that works. And so what I wanna do is I wanna take this first-degree term right over here, 11 x, and I wanna split it into a 12 x and a negative one x. So let's do that. So I can, let's just focus on this part, on this part right now and then I'll put it all back together at the end. So I can rewrite all of this business as four x squared. And instead of writing the 11 x there, I'm gonna use this blue color, I'm gonna break it up as a 12 x. So plus 12 x and then minus one x. Notice these two still add up to 11 x. And then I have my minus three. And then, let's see, out of these first two what can I factor out? Let's see, I can factor out a four x. So I can rewrite these first two as, and if this is unfamiliar to you I encourage you to review factoring by grouping on Khan Academy. So if we factor out a four x that's going to be, it's going to be, we're gonna be left with an x here. And then we're gonna be left with a three over here. And then these second two terms, if we factor out a negative one, so I'll write negative one, times, we're going to have an x plus three. And so then we can factor out the x plus three. So let's do that. I'm running out of colors. So I factor out the x plus three and I am left with x plus three times, times four x. And I'm gonna keep these colors the same so you know where I got 'em from. Four x minus one. This is a very colorful solution that we have over here. And there you have it. I factored the second part into these two factors. And so now I can put it all together. I can rewrite p of x as a product of linear factors. P of x is equal to x plus two times x plus three times four x minus one. Four x minus one. And we are done.