Dividing polynomials by linear expressions | Algebra 2 | Khan Academy

We're told to divide the polynomials. The form of your answer should either be just a clean polynomial or some polynomial plus some constant over x plus two, where p of x is a polynomial and k is an integer. Fair enough!

If we were doing this on Khan Academy, we would have to type this in, but we're just going to do it by hand. And like always, pause this video and try to do it on your own before we work through it together.

All right, now let's work through it together. What we're trying to do is divide x plus two into three x to the third power plus four x squared minus three x plus seven. Like always, we focus on the highest degree terms first. X goes into 3x to the third power how many times? Well, 3x squared times.

We'd want to put that in the second degree column: 3x squared. 3x squared times 2 is 6x squared. 3x squared times x is 3x to the third power. There's something very meditative about algebraic long division. Anyway, we'd want to subtract what we just wrote from what we have up here.

So let's subtract, and these characters cancel out. Then, 4x squared minus 6x squared is negative 2x squared. Bring down that negative 3x. Now we would want to say, "Hey, how many times does x go into negative 2x squared?" Well, it would go negative 2x times.

Put that in our first degree column: negative 2x. Negative 2x times 2 is negative 4x. Negative 2x times x is negative 2x squared. Now we want to subtract what we have here in orange from what we have up here in teal.

So we either put a negative around the whole thing or we distribute that negative. That becomes a positive, that becomes a positive, and so this is equal to the x squared terms cancel out. Negative 3x plus 4x is just going to be a straight-up x. Bring down that 7, x plus 7.

How many times does x go into x? Well, one time! Actually, let me use a new color here. So how many times does x go into x? It goes one time. Put that in the constant column. One times 2 is 2. One times x is x.

We want to subtract these characters, and we're left with 7 minus 2 is 5. So we can rewrite this whole thing as we deserve, I guess a little bit of a drum roll: three x squared minus two x plus one plus the remainder five over x plus two.

One way to think about is, "Hey, I have this remainder, I'd have to keep dividing it by x plus 2 if I really want to figure out exactly what this is." Now, if I wanted these expressions to be completely identical, I would put a condition on the domain that x cannot be equal to negative 2 because if x was equal to negative 2 we'd be dividing by 0 here.

But for the purposes of this exercise, you just have to input. You just have to input this part right over here. You'd have to type it in, which I guess isn't the easiest thing to do in the world, but it's worth doing. All right, see you in the next video!