Complex conjugate root theorem
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In mathematics, the complex conjugate root theorem states that if ''P'' is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
in one variable with
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
coefficients, and ''a'' + ''bi'' is a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of ''P'' with ''a'' and ''b'' real numbers, then its
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
''a'' − ''bi'' is also a root of ''P''. Preview available a
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/ref> It follows from this (and the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
) that, if the
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
of a real polynomial is
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
, it must have at least one real root. That fact can also be proved by using the intermediate value theorem.


Examples and consequences

* The polynomial ''x''2 + 1 = 0 has roots ± ''i''. * Any real square matrix of odd degree has at least one real
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
. For example, if the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
is orthogonal, then 1 or −1 is an eigenvalue. * The polynomial ::x^3 - 7x^2 + 41x - 87 :has roots ::3,\, 2 + 5i,\, 2 - 5i, :and thus can be factored as ::(x - 3)(x - 2 - 5i)(x - 2 + 5i). :In computing the product of the last two factors, the
imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s cancel, and we get ::(x - 3)(x^2 - 4x + 29). :The non-real factors come in pairs which when multiplied give
quadratic polynomial In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
s with real coefficients. Since every polynomial with
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
coefficients can be factored into 1st-degree factors (that is one way of stating the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
), it follows that every polynomial with real coefficients can be factored into factors of degree no higher than 2: just 1st-degree and quadratic factors. * If the roots are and , they form a quadratic ::x^2 - 2ax + (a^2 + b^2). : If the third root is , this becomes ::(x^2 - 2ax + (a^2 + b^2))(x-c) ::=x^3 + x^2(-2a-c) + x(2ac+a^2+b^2) - c(a^2 + b^2).


Corollary on odd-degree polynomials

It follows from the present theorem and the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
that if the degree of a real polynomial is odd, it must have at least one real root. This can be proved as follows. *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; *Therefore some of them must be real. This requires some care in the presence of multiple roots; but a complex root and its conjugate do have the same
multiplicity Multiplicity may refer to: In science and the humanities * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (philosophy), a philosophical concept * Multiplicity (psychology), having or using mult ...
(and this lemma is not hard to prove). It can also be worked around by considering only
irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted f ...
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 corollary can also be proved directly by using the intermediate value theorem.


Proof

One proof of the theorem is as follows: Consider the polynomial : P(z) = a_0 + a_1z + a_2z^2 + \cdots + a_nz^n where all ''a''''r'' are real. Suppose some
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
''ζ'' is a root of ''P'', that is P(\zeta) = 0. It needs to be shown that : P\big(\, \overline \,\big) = 0 as well. If ''P''(''ζ''  ) = 0, then : a_0 + a_1\zeta + a_2\zeta^2 + \cdots + a_n\zeta^n = 0 which can be put as : \sum_^n a_r\zeta^r = 0. Now : P\big(\, \overline \,\big) = \sum_^n a_r \big(\, \overline \,\big)^r and given the properties of complex conjugation, : \sum_^n a_r\big(\, \overline \,\big)^r = \sum_^n a_r \overline = \sum_^n \overline = \overline. Since : \overline = \overline, it follows that : \sum_^n a_r\big(\, \overline \,\big)^r = \overline = 0. That is, : P\big(\, \overline \,\big) = a_0 + a_1\overline + a_2\big(\, \overline \,\big)^2 + \cdots + a_n\big(\, \overline \,\big)^n = 0. Note that this works only because the ''a''''r'' are real, that is, \overline = a_r. If any of the coefficients were non-real, the roots would not necessarily come in conjugate pairs.


Notes

{{Reflist Theorems in complex analysis Theorems about polynomials Articles containing proofs