Division in Polynomial Rings

Haven · #1 $Old$ November 4th, 2009, 01:29 AM

I'm having troubles with this problem

Let $F$ be a Field, and suppose that $a\in F$ and $f(x)\in F[x]$ . Show $f(a)$ is the remainder when $f(x)$ is divided by $x-a$

The case when $f(a)=0$ is trivial but I don't know what to do from there.

Opalg · #2 $Old$ November 4th, 2009, 02:35 AM

Quote:

Originally Posted by Haven $View Post$

I'm having troubles with this problem

Let $F$ be a Field, and suppose that $a\in F$ and $f(x)\in F[x]$ . Show $f(a)$ is the remainder when $f(x)$ is divided by $x-a$

The case when $f(a)=0$ is trivial but I don't know what to do from there.

If q(x) is the quotient and r is the remainder when f(x) is divided by x–a then, by definition, f(x) = (x–a)q(x) + r. Now all you have to do is to put x=a.

Haven · #3 $Old$ November 4th, 2009, 11:20 AM

That is so simple, I can't believe I didn't get that.