# Math Help - Polynomials

1. ## Polynomials

Please help me with this proof! Thank you!

Let F be a field. Let f(x), g(x) exist in F[x] be monic polynomials such that f(x)|g(x)
and deg g(x) is less than or equal to deg f(x). Prove that f(x) = g(x).

2. Originally Posted by mathgirl1188
Please help me with this proof! Thank you!

Let F be a field. Let f(x), g(x) exist in F[x] be monic polynomials such that f(x)|g(x)
and deg g(x) is less than or equal to deg f(x). Prove that f(x) = g(x).

$f(x)\mid g(x)\Longrightarrow g(x)=f(x)h(x)\Longrightarrow deg(g)=deg(f)+deg(h)$ (why?) , but we're given that

$deg(x)\leq deg(f)\Longrightarrow deg(f)+deg(h)\leq deg(f)\Longrightarrow deg(h)=0$ (why?), and comparing

superior coefficients in $g(x)=f(x)h(x)$ we solve the problem.

Tonio