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Theorem — The number of strictly positive roots (counting multiplicity) of is equal to the number of sign changes in the coefficients of , minus a nonnegative even number. If b 0 > 0 {\displaystyle b_{0}>0} , then we can divide the polynomial by x b 0 {\displaystyle x^{b_{0}}} , which would not change its number of strictly positive roots.
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;
Root (or zero) of a polynomial: Given a polynomial p(x), the x values that satisfy p(x) = 0 are called roots (or zeroes) of the polynomial p. Graphing End behaviour –
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.
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.
According to the French school of mathematics of the 19th century, this is the first step in computing the real roots, the second being their approximation to any degree of accuracy; moreover, the focus is on the positive roots, because to isolate the negative roots of the polynomial p(x) replace x by −x (x ← −x) and repeat the process.
Linear congruence theorem (number theory, modular arithmetic) Linear speedup theorem (computational complexity theory) Linnik's theorem (number theory) Lions–Lax–Milgram theorem (partial differential equations) Liouville's theorem (complex analysis, entire functions) Liouville's theorem (conformal mappings) Liouville's theorem (Hamiltonian ...
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