Factoring using polynomial division | Algebra 2 | Khan Academy

We are told the polynomial p of x is equal to 4x to the third plus 19x squared plus 19x minus 6 has a known factor of x plus 2. 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 has a known factor of x plus 2, this polynomial would not be so easy to factor. But because we know we have a known factor of x plus 2, 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 2 into our polynomial. So it's 4x to the third power plus 19x squared plus 19x minus 6. And we've done this multiple times already. We look at the highest degree terms.

x goes into 4x to the third 4x squared times. I put that in the x squared or the second degree column. 4x squared times x is 4x to the third. 4x squared times 2 is 8x. So 8x squared.

Then I want to subtract these from what I have up here. So I'll subtract, and then I'm going to be left with 19x squared minus 8x squared, which is 11x squared. Then I will bring down the 19x, so plus 19x.

And so once again, I look at x and 11x squared. x goes into 11x squared 11x times, so that's plus 11x. 11x times x is 11x squared. 11x times 2 is 22x.

I need to subtract these from what we have in that teal color, and we are left with 19 minus 22 of something is negative 3 of that something. In this case, it's negative 3x.

Then we bring down that negative 6, and then we look at once again at the x and the negative 3x. x goes into negative 3x negative 3 times, and so negative 3 times x is negative 3x. Negative 3 times 2 is negative 6.

Then we want to 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.

So we can rewrite p of x now. We can rewrite p of x as being equal to x plus 2 times all of this business: 4x squared plus 11x minus 3. 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 4x squared plus 11x minus 3, that's still quadratic, so we have to factor that further. And let's see, there are a couple of ways we could approach it. We could use, well, we could try something like the quadratic formula, or we could factor by grouping.

To factor by grouping, the whole reason we have to factor by grouping is we have a leading coefficient here that is not one. So we need to think of two numbers whose product is equal to four times negative three.

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 twelve, and a plus b needs to be equal to eleven. So the best that I can think of, they have to be opposite signs because their product is a negative.

So if I had positive 12 and negative 1, that works. If a is equal to negative 1 and b is equal to positive 12, that works. And so what I want to do is I want to take this first degree term right over here, 11x, and I want to split it into a 12x and a negative 1x.

So let's do that. I can, let's just focus 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 4x squared, and instead of writing the 11x there, I'm going to use this blue color. I'm going to break it up as a 12x.

So plus 12x and then minus 1x. Notice these two still add up to 11x, and then I have my minus 3. Then let's see, out of these first two, what can I factor out? Let's see, I can factor out a 4x.

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 4x, that's going to be... we're going to be left with an x here, and then we're going to be left with a 3 over here.

And then these second two terms, if we factor out a negative one, so I'll write negative one times... we're gonna have an x plus 3. And so then we can factor out the x plus 3. So let's do that.

I'm running out of colors, so I factor out the x plus 3, and I am left with x plus 3 times... I’m going to keep these colors the same so you know where I got them from: 4x minus 1. 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 2 times x plus 3 times 4x minus 1. 4x minus 1, and we are done.