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The FOIL rule converts a product of two binomials into a sum of four (or fewer, if like terms are then combined) monomials. [6] The reverse process is called factoring or factorization. In particular, if the proof above is read in reverse it illustrates the technique called factoring by grouping.
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. 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.
The question of when this happens is rather subtle: for example, for the localization of k[x, y, z]/(x 2 + y 3 + z 5) at the prime ideal (x, y, z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x, y, z]/(x 2 + y 3 + z 7) at the prime ideal (x, y, z) the local ring is a UFD but ...
A product of monic polynomials is monic. A product of polynomials is monic if and only if the product of the leading coefficients of the factors equals 1. This implies that, the monic polynomials in a univariate polynomial ring over a commutative ring form a monoid under polynomial multiplication.
A primitive monomial is a special case of a monomial in this second sense, where the coefficient is . For example, in this interpretation − 7 x 5 {\displaystyle -7x^{5}} and ( 3 − 4 i ) x 4 y z 13 {\displaystyle (3-4i)x^{4}yz^{13}} are monomials (in the second example, the variables are x , y , z , {\displaystyle x,y,z,} and the coefficient ...
If one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form (x − r)Q(x) where Q(x) is a polynomial of degree n − 1. Q ( x ) is simply the quotient obtained from the division process; since r is known to be a root of P ( x ), it is known that the remainder must be zero.