A polynomial 𝑓 of 𝑥 is divided by 𝑥 minus 𝑎. Given that 𝑥 minus 𝑎 is not a factor of 𝑓 of 𝑥, what is the remainder equal to?
Well, there is a theorem called the polynomial remainder theorem, or just the remainder theorem, that says that the remainder when dividing a polynomial 𝑓 of 𝑥 by the linear polynomial 𝑥 minus 𝑎 is 𝑓 of 𝑎. So using the theorem, we see that the answer is just 𝑓 of 𝑎. This theorem is very useful in practice because it allows us to find the remainder without going through the whole process of long division.
We can prove this theorem by using what we know about the polynomial long division. When we use polynomial long division to divide a polynomial 𝑓 of 𝑥 by 𝑥 minus 𝑎, essentially what we’re doing is rewriting it in the form 𝑥 minus 𝑎 times 𝑞 of 𝑥, the quotient, plus 𝑟 of 𝑥, the remainder.
In general, this remainder, 𝑟 of 𝑥, is a polynomial in 𝑥. However, we know that the degree of the polynomial 𝑟 of 𝑥 is always less than the degree of the polynomial that we’re dividing by. In this case, 𝑥 minus 𝑎. So we know that 𝑟 of 𝑥 is just a constant polynomial, and we’ll write it just as 𝑟.
Okay. So we know that the remainder is a constant. I suppose that’s slightly helpful. But we still need to know what that constant is. This equality holds for all values of 𝑥. So we can substitute whatever value of 𝑥 we’d like in there, and it will still be true. We can choose to substitute in 𝑎, and we’ll get that 𝑓 of 𝑎 is equal to 𝑎 minus 𝑎 times 𝑞 of 𝑎 plus 𝑟. And of course, 𝑎 minus 𝑎 is just zero. So we get zero 𝑞 of 𝑎 plus 𝑟 which is just 𝑟.
So that’s the proof of the theorem that we’ve just used. And we can see that it works even if 𝑥 minus 𝑎 is a factor of 𝑓 of 𝑥, in which case 𝑟 will just be zero. So when a polynomial 𝑓 of 𝑥 is divided by 𝑥 minus 𝑎, given that 𝑥 minus 𝑎 is not a factor of 𝑓 of 𝑥 and even when it is in fact, the remainder is equal to 𝑓 of 𝑎.