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Math Help - Complex Roots

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by Coda202 View Post
    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 f(x) = a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0 we are told that f(\alpha) = 0 therefore a_n\alpha^n + ... + a_1\alpha + a_0 = 0 where \alpha = a+bi.

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