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In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial ...
For example, the fundamental theorem of algebra, which states that every polynomial with complex coefficients has complex roots, implies that a polynomial with integer coefficients can be factored (with root-finding algorithms) into linear factors over the complex field C.
The formula for the difference of two squares can be used for factoring polynomials that contain the square of a first quantity minus the square of a second quantity. For example, the polynomial x 4 − 1 {\displaystyle x^{4}-1} can be factored as follows:
Given a quadratic polynomial of the form + + it is possible to factor out the coefficient a, and then complete the square for the resulting monic polynomial. Example: + + = [+ +] = [(+) +] = (+) + = (+) + This process of factoring out the coefficient a can further be simplified by only factorising it out of the first 2 terms. The integer at the ...
In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if f ( x ) {\displaystyle f(x)} is a polynomial, then x − a {\displaystyle x-a} is a factor of f ( x ) {\displaystyle f(x)} if and only if f ( a ) = 0 {\displaystyle f(a)=0} (that is, a {\displaystyle a} is a root of the polynomial).
Polynomial long division can be used to find the equation of the line that is tangent to the graph of the function defined by the polynomial P(x) at a particular point x = r. [3] If R ( x ) is the remainder of the division of P ( x ) by ( x – r ) 2 , then the equation of the tangent line at x = r to the graph of the function y = P ( x ) is y ...
So, except for very low degrees, root finding of polynomials consists of finding approximations of the roots. By the fundamental theorem of algebra, a polynomial of degree n has exactly n real or complex roots counting multiplicities. It follows that the problem of root finding for polynomials may be split in three different subproblems;
In particular, the unique factorization property of polynomials can be stated as: Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials. Vieta's formulas are simpler in the case of monic polynomials: The i th elementary symmetric function of the roots of a monic ...
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