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
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.