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In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property , a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials .
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds.
It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root x = r is found, a linear polynomial ( x – r ) can be factored out of the polynomial using polynomial long division , resulting in a polynomial of lower degree whose roots are also roots of the original polynomial.
In algebra, Gauss's lemma, [1] named after Carl Friedrich Gauss, is a theorem [note 1] about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic).
The terminology used to describe algebraic fractions is similar to that used for ordinary fractions. For example, an algebraic fraction is in lowest terms if the only factors common to the numerator and the denominator are 1 and −1. An algebraic fraction whose numerator or denominator, or both, contain a fraction, such as + , is called ...