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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
 #1
avatar+6046 
+1
Best Answer

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

Rom May 19, 2019
edited by Rom  May 20, 2019

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