1. ## Complex Roots

Let f be in R[x] and suppose a+bi is a complex root of f with a,b in R and b not equal to 0. Prove that a-bi is also a root of f.

I let h = (x-(a+bi))(x-(a-bi)). I know I need to show h is in R[x] in order to apply the division algorithm to f and h in R[x], but I'm stumped on how to do that.

2. Originally Posted by Coda202
Let f be in R[x] and suppose a+bi is a complex root of f with a,b in R and b not equal to 0. Prove that a-bi is also a root of f.

I let h = (x-(a+bi))(x-(a-bi)). I know I need to show h is in R[x] in order to apply the division algorithm to f and h in R[x], but I'm stumped on how to do that.
Let $\displaystyle f(x) = a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0$ we are told that $\displaystyle f(\alpha) = 0$ therefore $\displaystyle a_n\alpha^n + ... + a_1\alpha + a_0 = 0$ where $\displaystyle \alpha = a+bi$.

This means, $\displaystyle \overline{a_n\alpha^n + ... + a_1\alpha +a_0} = 0 \implies a_n \beta^n + ... + a_1\beta + a_0 = 0$ where $\displaystyle \beta = \overline{\alpha} = a-bi$.