How do I should that $x-1$ a factor of a positive degree polynomial?

Question

I'm supposed to show that $x-1$ is a factor of a polynomial P of positive degree if and only if the sum of the coefficients of P is zero. How do I do that exactly?

Answer 1

If $x-1$ is a factor then you have $p(x) = (x-1)q(x)$ where q(x) $ is another polynomial.

Let $x=1$, we get $$p(1) = (1-1)q(1)=0$$

That is if you let $x=1$ in your polynomial you will get $p(1)=0$

Note that when you find $p(1)$ you let $x=1$ , so you are just adding the coefficients of $P(x)$ together.

Thus if $x=1$ is a factor, the sum of coefficients of $p(x)$ is $0$.

Answer 2

The remainder of the division of a polynomial $p(x)$ by $x-a$ is $p(a)$, so $$p(x)\;\text{ divisible by }\;x-a\iff p(a)=0.$$ On the other hand $p(1)$ is just the sum of the coefficients of $p(x)$, whence the conclusion.