Revision as of 14:27, 1 January 2014

Theorem

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.

Uses

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

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Difference between revisions of "Conjugate Root Theorem"

Revision as of 14:27, 1 January 2014

Theorem

Uses