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

Mathrules May 19, 2019

#1**+1 **

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

.Rom May 19, 2019

#1**+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