Complex conjugate root theorem

Apr 24, 2026 - 07:51
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Complex conjugate root theorem

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{{Short description|Theorem about polynomials}}
{{Short description|Theorem about polynomials}}
In [[mathematics]], the '''complex conjugate root theorem''' states that if ''P'' is a [[polynomial]] in one variable with [[real number|real]] [[coefficient]]s, and ''a'' + ''bi'' is a [[root of a polynomial|root]] of ''P'' with ''a'' and ''b'' being real numbers, then its [[complex conjugate]] ''a'' − ''bi'' is also a root of ''P''.{{cite book|title=Maynooth Mathematical Olympiad Manual|author=Anthony G. O'Farell and Gary McGuire|pages=104|chapter=Complex numbers, 8.4.2 Complex roots of real polynomials|year=2002|publisher=Logic Press |isbn=0954426908}} Preview available at [https://books.google.com/books?q=Maynooth+Mathematical+Olympiad+Manual&ots=xQ0hpAQkpc&sa=X&oi=print&ct=title Google books]
In [[mathematics]], the '''complex conjugate root theorem''' states that if ''P'' is a [[polynomial]] in one variable with [[real number|real]] [[coefficient]]s, and ''a'' + ''bi'' is a [[root of a polynomial|root]] of ''P'' with ''a'' and ''b'' being [[Real number|real numbers]], then its [[complex conjugate]] ''a'' − ''bi'' is also a root of ''P''.{{cite book|title=Maynooth Mathematical Olympiad Manual|author=Anthony G. O'Farell and Gary McGuire|pages=104|chapter=Complex numbers, 8.4.2 Complex roots of real polynomials|year=2002|publisher=Logic Press |isbn=0954426908}} Preview available at [https://books.google.com/books?q=Maynooth+Mathematical+Olympiad+Manual&ots=xQ0hpAQkpc&sa=X&oi=print&ct=title Google books]


It follows from this (and the [[fundamental theorem of algebra]]) that, if the [[degree of a polynomial|degree]] of a real polynomial is [[parity (mathematics)|odd]], it must have at least one real root.{{cite book|title=Complex Analysis and Applications|author=Alan Jeffrey|chapter=Analytic Functions|pages=22–23|year=2005|publisher=CRC Press|isbn=158488553X}} That fact can also be [[mathematical proof|proved]] by using the [[intermediate value theorem]].
It follows from this (and the [[fundamental theorem of algebra]]) that, if the [[degree of a polynomial|degree]] of a real [[polynomial]] is [[parity (mathematics)|odd]], it must have at least one real root.{{cite book|title=Complex Analysis and Applications|author=Alan Jeffrey|chapter=Analytic Functions|pages=22–23|year=2005|publisher=CRC Press|isbn=158488553X}} That fact can also be [[mathematical proof|proved]] by using the [[intermediate value theorem]].


== Examples and consequences ==
== Examples and consequences ==
* The polynomial ''x''<sup>2 + 1 = 0 has roots ±''i''.
*The polynomial <math>x^2 + 1 = 0 has two imaginary roots: \pm i

* Any real [[square matrix]] of odd degree has at least one real [[eigenvalue]]. For example, if the [[matrix (mathematics)|matrix]] is [[orthogonal matrix|orthogonal]], then 1 or −1 is an eigenvalue.
* Any real [[square matrix]] of odd degree has at least one real [[eigenvalue]]. For example, if the [[matrix (mathematics)|matrix]] is [[orthogonal matrix|orthogonal]], then 1 or −1 is an eigenvalue.
* The polynomial
* The polynomial
::x^3 - 7x^2 + 41x - 87
::x^3 - 7x^2 + 41x - 87
:has roots
:has roots of
::3,\, 2 + 5i,\, 2 - 5i,
::3,\, 2 + 5i,\, 2 - 5i,
:and thus can be factored as
:and thus can be factored as
::(x - 3)(x - 2 - 5i)(x - 2 + 5i).
::(x - 3)(x - 2 - 5i)(x - 2 + 5i).
:In computing the product of the last two factors, the [[imaginary part]]s cancel, and we get
:In computing the product of the last two factors, the [[imaginary part]]s cancel, and we get
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*Since non-real complex roots come in conjugate pairs, there are an [[even number]] of them;
*Since non-real complex roots come in conjugate pairs, there are an [[even number]] of them;
*But a polynomial of odd degree has an odd number of roots ([[fundamental theorem of algebra]]);
*But a polynomial of odd degree has an odd number of roots ([[fundamental theorem of algebra]]);
*Therefore some of them must be real.
*Therefore, some of them must be real.
This requires some care in the presence of [[multiple root]]s; but a complex root and its conjugate do have the same [[Multiplicity (mathematics)|multiplicity]] (and this [[lemma (mathematics)|lemma]] is not hard to prove). It can also be worked around by considering only [[irreducible polynomial]]s; any real polynomial of odd degree must have an irreducible factor of odd degree, which (having no multiple roots) must have a real root by the reasoning above.
This requires some care in the presence of [[multiple root]]s; but a complex root and its conjugate do have the same [[Multiplicity (mathematics)|multiplicity]] (and this [[lemma (mathematics)|lemma]] is not hard to prove). It can also be worked around by considering only [[irreducible polynomial]]s; any real polynomial of odd degree must have an irreducible factor of odd degree, which (having no multiple roots) must have a real root by the reasoning above.
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