Editing Polynomial long division (section)

===Factoring polynomials===

Sometimes one or more roots of a polynomial are known, perhaps having been found using the [[rational root theorem]]. 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 {{nowrap|(''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.

Likewise, if several roots ''r'', ''s'', . . .  of ''P''(''x'') are known, a linear factor {{nowrap|(''x'' − ''r'')}} can be divided out to obtain ''Q''(''x''), and then {{nowrap|(''x'' − ''s'')}} can be divided out of ''Q''(''x''), etc. Alternatively, the quadratic factor <math>(x-r)(x-s)=x^2-(r{+}s)x+rs</math> can be divided out of ''P''(''x'') to obtain a quotient of degree {{nowrap|''n'' − 2.}}

This method is especially useful for cubic polynomials, and sometimes all the roots of a higher-degree polynomial can be obtained. For example, if the rational root theorem produces a single (rational) root of a [[quintic function|quintic polynomial]], it can be factored out to obtain a quartic (fourth degree) quotient; the explicit formula for the roots of a [[quartic function|quartic polynomial]] can then be used to find the other four roots of the quintic. There is, however, no general way to solve a quintic by purely algebraic methods, see [[Abel–Ruffini theorem]].