Dividing polynomials by linear expressions: missing term | Algebra 2 | Khan Academy

In front of us, we have another screenshot from Khan Academy, and I've modified a little bit so I have a little bit of extra space. It says, "Divide the polynomials. The form of your answer should either be a straight-up polynomial or a polynomial plus the remainder over x minus 5," which we have here in the denominator where P of x is a polynomial and K is an integer.

So we've done stuff like this, but like always, I encourage you to pause this video and work on this on your own. If you're doing this on Khan Academy, there's a little bit of an input box here where you'd have to type in the answer. But let's just do it on paper for now.

All right, so we're trying to figure out what x minus 5 divided into 2x to the third power. And actually, I want to be careful here because I want to be very, very organized about my different degree columns. So this is my third-degree column, and then I want my second-degree column. But there is no second-degree term here; there's a first-degree term, so I'll write it out here: minus 47x.

And actually, to be even more careful, I'll write plus 0x squared, and then I have minus 15. By putting that plus 0x squared, that's making sure I'm doing good, I guess, degree place column hygiene.

All right, so now we can work through this. First, we could say, "Hey, how many times does x go into the highest degree term here?" Well, x goes into 2x to the third power 2x squared times, and we'd want to put that in the second-degree column: 2x squared. You can see how it would have gotten messy if I put negative 47x here. I’d be like, "Where do I put that 2x squared?" And you might have confused yourself, which none of us would want to happen.

All right, 2x squared times negative 5 is negative 10x squared. 2x squared times x is 2x to the third power. Now we want to subtract what we have in red from what we have in blue. So I'll multiply them both by negative 1, so that becomes a negative and then that one becomes a positive. That's actually one of the biggest areas for careless errors: if you have negative here and you just want to subtract it, because you know you have to subtract, be like, "No, I'm subtracting a negative; it needs to be a positive now."

All right, so 0x squared plus 10x squared is 10x squared, and then the 2x to the third minus 2x to the third is just 0. And then we can bring down that negative 47x. Once again, we look at the highest degree terms. x goes into 10x squared 10x times, so plus 10x. 10x times negative 5 is negative 50x.

Negative 50x, 10x times x is 10x squared. And once again, we want to subtract what we have in teal from what we have in red. So we can multiply both of these times negative 1; that becomes a negative, this one becomes a positive. Now, negative 47x plus 50x is positive 3x, and then 10x squared minus 10x squared gets canceled out. Bring down that 15.

Come on down! I used to watch a lot of Prices Right growing up, never quite made it to the show. All right, x goes into 3x how many times? It goes three times. 3 times negative 5 is negative 15. 3 times x is 3x. We want to subtract the orange from the teal, and so this becomes a negative, this becomes a positive.

15, my or negative 15 plus 15 is a 0, and 3x minus 3x is 0. So you're just left with 0. So no remainder. So this whole thing you could re-express or simplify as 2x squared plus 10x plus 3.

And once again, if this was on Khan Academy, there'd be a little bit of an input box that looks something like this, and you would have to type this in. Now, if you wanted these to be exactly the same expression, you would also need to constrain the domain. You would say, "Okay, for x does not equal positive 5."

And the reason we have to constrain that is where the whole, the whole reason why we can even divide by x minus five is we're assuming that x minus five is not equal to zero. And it's generally not equal to zero as long as x does not equal to positive five. But for the sake of that exercise, you don't need to constrain the domain like this.