Difference between revisions of "Complex Conjugate Root Theorem"
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− | + | In [[algebra]], the '''complex conjugate root theorem''' states that if <math>P(x)</math> is a [[polynomial]] with [[real number | real coefficients]], then a [[complex number]] is a root of <math>P(x)</math> if and only if its [[complex conjugate]] is also a root. | |
+ | |||
+ | A common intermediate step is to present a complex root of a real polynomial without its conjugate. It is then up to the solver to recognize that its conjugate is also a root. | ||
+ | |||
+ | == Proof == | ||
+ | Let <math>P(x)</math> have the form <math>a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0</math>, where constants <math>a_n, a_{n-1}, \ldots, a_1, a_0</math> are real numbers, and let <math>z</math> be a complex root of <math>P(x)</math>. We then wish to show that <math>\overline{z}</math>, the complex conjugate of <math>z</math>, is also a root of <math>P(x)</math>. Because <math>z</math> is a root of <math>P(x)</math>, <cmath>P(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0 = 0.</cmath> Then by the [https://artofproblemsolving.com/wiki/index.php/Complex_conjugate#Properties properties of complex conjugation], | ||
+ | <cmath>\begin{align*} | ||
+ | \overline{a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0} = \overline{0} \\ | ||
+ | \overline{a_n z^n} + \overline{a_{n-1} z^{n-1}} + \cdots + \overline{a_1 z} + \overline{a_0} = 0 \\ | ||
+ | a_n \overline{z^n} + a_{n-1} \overline{z^{n-1}} + \cdots + a_1 \overline{z} + a_0 = 0 \\ | ||
+ | a_n \overline{z}^n + a_{n-1} \overline{z}^{n-1} + \cdots + a_1 \overline{z} + a_0 = 0 \\ | ||
+ | P(\overline{z}) = 0, | ||
+ | \end{align*}</cmath> | ||
+ | which entails that <math>\overline{z}</math> is a root of <math>P(x)</math>, as required. <math>\square</math> | ||
+ | |||
+ | == See also == | ||
+ | * [[Polynomial]] | ||
+ | |||
+ | [[Category:Algebra]] | ||
+ | [[Category:Polynomials]] | ||
+ | [[Category:Theorems]] |
Revision as of 20:18, 27 November 2021
In algebra, the complex conjugate root theorem states that if is a polynomial with real coefficients, then a complex number is a root of if and only if its complex conjugate is also a root.
A common intermediate step is to present a complex root of a real polynomial without its conjugate. It is then up to the solver to recognize that its conjugate is also a root.
Proof
Let have the form , where constants are real numbers, and let be a complex root of . We then wish to show that , the complex conjugate of , is also a root of . Because is a root of , Then by the properties of complex conjugation, which entails that is a root of , as required.