Conjugate Root Theorem

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The Conjugate Root Theorem states that if $P(x)$ is a polynomial with real coefficients, and $a+bi$ is a root of the equation $P(x) = 0$, where $i = \sqrt{-1}$, then $a-bi$ is also a root. A similar theorem states that if $P(x)$ is a polynomial with rational coefficients and $a+b\sqrt{c}$ is a root of the polynomial, then $a-b\sqrt{c}$ is also a root.


Suppose that $P(a + bi) = 0$. Then $\overline{P(a + bi)} = 0$. However, we know that $\overline{P(a + bi)} = \overline{P}(\overline{a + bi}) = P(a - bi)$, where we define $\overline{P}$ to be the polynomial with the coefficients replaced with their complex conjugates; we know that $\overline{P} = P$ by the assumption that $P$ has real coefficients. Thusly, we show that $P(a - bi) = 0$, and we are done.


This has many uses. If you get a fourth degree polynomial, and you are given that a number in the form of $a+bi$ is a root, then you know that $a-bi$ in the root. Using the Factor Theorem, you know that $(x-(a+bi))(x-(a-bi))$ is also a root. Thus, you can multiply that out, and divide it by the original polynomial, to get a depressed quadratic equation. Of course, it doesn't have to be a fourth degree polynomial. It could just simplify it a bit.

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