Dividing polynomials by linear factors
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- We're told, 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, and if we were doing this on Khan Academy, this is a screen shot from 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. And 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. And so, like always, we focus on the highest degree terms first. x goes into three x to the third power how many times? Three x squared times. We'd want to put that in the second degree column. Three x squared. Three x squared times two is six x squared. Three x squared times x is three x 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. And then four x squared minus six x squared is negative two x squared. Bring down that negative three x. And now we would wanna say, hey, how many times does x go into negative two x squared? Well, it would go negative two x times. Put that in our first degree column. Negative two x times two is negative four x. Negative two x times x is negative two x squared. Now we wanna 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 and that becomes a positive, that becomes a positive. And so this is equal to, the x squared terms cancel out. Negative three x plus four x is just going to be a straight up x. Bring down that seven, x plus seven. How many times does x go into x? Well, one time. Actually, let me do a, 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 two is two. One times x is x. We wanna subtract these characters. And we're left with seven minus two is five. And 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 it is, hey, I have this remainder, I'd have to keep dividing it by x plus two if I really wanted 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 two because if x was equal to negative two, we'd be dividing by zero here. But for the purposes of this exercise, 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.