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Suppose f(x) is a polynomial of degree nwith coefficients in C. Consider the fo…

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Wize University Linear Algebra Textbook > Complex Numbers

Powers and Roots

5 Activities

Suppose f(x)f(x) is a polynomial of degree nnwith coefficients in C\mathbb{C}. Consider the following statements:
(i) If z=a+biz = a + bi is a root of f(x)f(x), then so is z=a−biz = a - bi.
(ii) If n=2n = 2 and f(x)f(x) has a nonzero linear term, then f(x)f(x) can only have roots if its coefficients are real.
(iii) If f(x)=zn−kf(x) = z^n - k where k∈C\{0}k\in\mathbb{C}\backslash\{0\}, then f(x)f(x) can be factored into nn distinct linear factors over C\mathbb{C}.
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