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Ruffini's rule
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==Application to polynomial factorization== Ruffini's rule can be used when one needs the quotient of a polynomial {{mvar|P}} by a binomial of the form <math>x-r.</math> (When one needs only the remainder, the [[polynomial remainder theorem]] provides a simpler method.) A typical example, where one needs the quotient, is the [[polynomial factorization|factorization]] of a polynomial <math>p(x)</math> for which one knows a root {{mvar|r}}: The remainder of the [[Euclidean division]] of <math>p(x)</math> by {{mvar|r}} is {{math|0}}, and, if the quotient is <math>q(x),</math> the Euclidean division is written as :<math>p(x)=q(x)\,(x-r).</math> This gives a (possibly partial) factorization of <math>p(x),</math> which can be computed with Ruffini's rule. Then, <math>p(x)</math> can be further factored by factoring <math>q(x).</math> The [[fundamental theorem of algebra]] states that every polynomial of positive degree has at least one [[complex number|complex]] root. The above process shows the fundamental theorem of algebra implies that every polynomial {{math|1=''p''(''x'') = ''a''<sub>''n''</sub>''x''<sup>''n''</sup> + ''a''<sub>''n''β1</sub>''x''<sup>''n''β1</sup> + β― + ''a''<sub>1</sub>''x'' + ''a''<sub>0</sub>}} can be factored as :<math>p(x)=a_n(x-r_1)\cdots(x-r_n),</math> where <math>r_1,\ldots,r_n</math> are complex numbers.
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