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It follows from the present theorem and the fundamental theorem of algebra that if the degree of a real polynomial is odd, it must have at least one real root. [2] This can be proved as follows. Since non-real complex roots come in conjugate pairs, there are an even number of them;
The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is a n = 1.
Manin–Drinfeld theorem (number theory) Mann's theorem (number theory) Marcinkiewicz theorem (functional analysis) Marden's theorem (polynomials) Mazur's control theorem (number theory) Mergelyan's theorem (complex analysis) Marginal value theorem (biology, optimization) Markus−Yamabe theorem (dynamical systems)
Sturm's theorem expresses the number of distinct real roots of p located in an interval in terms of the number of changes of signs of the values of the Sturm sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of p .
The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real roots appear in pairs of the form (a + ib, a – ib).. It follows that the roots of a polynomial with real coefficients are mirror-symmetric with respect to the real axis.
Given any polynomial p, a root of p is a number y such that p(y) = 0. For example, the n th roots of x are the roots of the polynomial (in y) . Abel–Ruffini theorem states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of n th roots.
Vieta's formulas can be proved by considering the equality + + + + = () (which is true since ,, …, are all the roots of this polynomial), expanding the products in the right-hand side, and equating the coefficients of each power of between the two members of the equation.
Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting). For finding one root, Newton's method and other general iterative methods work generally well.
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