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explain why a polynomial equation with real coefficients and root 1-i must be of degree two or greater

May 19, 2019

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$$\text{Suppose p(x) is degree 1 or less and has 1-i as a root}\\ c_0+c_1(1-i) = 0\\ c_0+c_1 = i c_1\\ \text{The left side is the sum of two real numbers and thus real.\\ The right side is the product of a real number and i and is thus \\ imaginary so they cannot be equal..\\ Thus p(x) must be of degree 2 or greater}$$

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May 19, 2019
edited by Rom  May 20, 2019

$$\text{Suppose p(x) is degree 1 or less and has 1-i as a root}\\ c_0+c_1(1-i) = 0\\ c_0+c_1 = i c_1\\ \text{The left side is the sum of two real numbers and thus real.\\ The right side is the product of a real number and i and is thus \\ imaginary so they cannot be equal..\\ Thus p(x) must be of degree 2 or greater}$$