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Complex conjugate root theorem
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the complex conjugate root theorem states that if <i>P</i> is a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> in one variable with <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a>, and <i>a</i> + <i>bi</i> is a <a href="/facts/Root_of_a_polynomial/mlK3dhoT">root</a> of <i>P</i> with <i>a</i> and <i>b</i> being real numbers, then its <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> <i>a</i> − <i>bi</i> is also a root of <i>P</i>. </p><p>It follows from this (and the <a href="/facts/Fundamental_theorem_of_algebra/vspflSsM">fundamental theorem of algebra</a>) that, if the <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> of a real polynomial is <a href="/facts/Parity_(mathematics)/7zq2UM4V">odd</a>, it must have at least one real root. That fact can also be <a href="/facts/Mathematical_proof/7ECDrU80">proved</a> by using the <a href="/facts/Intermediate_value_theorem/SaFdYE7W">intermediate value theorem</a>. </p>