Editing Polynomial long division (section)

==Applications==

===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]].

===Finding tangents to polynomial functions===

Polynomial long division can be used to find the equation of the line that is [[tangent]] to the [[graph of a function|graph of the function]] defined by the polynomial ''P''(''x'') at a particular point {{nowrap|''x'' {{=}} ''r''.}}<ref>Strickland-Constable, Charles, "A simple method for finding tangents to polynomial graphs", ''[[Mathematical Gazette]]'' 89, November 2005: 466-467.</ref> If ''R''(''x'') is the remainder of the division of ''P''(''x'') by {{nowrap|(''x'' − ''r'')<sup>2</sup>,}} then the equation of the tangent line at {{nowrap|''x'' {{=}} ''r''}} to the graph of the function {{nowrap|''y'' {{=}} ''P''(''x'')}} is {{nowrap|''y'' {{=}} ''R''(''x''),}} regardless of whether or not ''r''  is a root of the polynomial.

====Example====
Find the equation of the line that is tangent to the following curve
<math>y = (x^3 - 12x^2 - 42)  </math>
:at: <math>x = 1 </math>
Begin by dividing the polynomial by:
<math> (x-1)^2=(x^2-2x+1)</math>

: <math>
\begin{array}{r}
 x - 10\\
 x^2-2x+1\ \overline{)\ x^3 - 12x^2 + 0x - 42}\\
 \underline{x^3 - {\color{White}0}2x^2 + {\color{White}1}x} {\color{White} {} - 42}\\
 -10x^2 - {\color{White}01}x - 42\\
 \underline{-10x^2 + 20x - 10}\\
 -21x - 32
\end{array}
</math>
The tangent line is 
<math> y=(-21x-32)</math>

===Cyclic redundancy check===

A [[cyclic redundancy check]] uses the remainder of polynomial division to detect errors in transmitted messages.