Polynomial remainder theorem
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The polynomial remainder theorem in algebra is an application of polynomial long division. It states that for polynomial <math>f(x)</math> that is divided by a linear divisor <math>x-a</math>, the remainder <math>r</math> is equal to <math>f(a)</math>.
This can be demonstrated by the definition of polynomial long division:
- <math>\frac{f(x)}{g(x)}=q(x) + \frac{r}{g(x)}</math>
Set the divisor <math>g(x)</math> to the linear divisor <math>x-a</math>:
- <math>\frac{f(x)}{x-a}=q(x) + \frac{r}{x-a}</math>
Solve for <math>f(x)</math>:
- <math>\frac{}{}f(x)=q(x)(x-a) + r</math>
Set <math>x=a</math>:
- <math>\frac{}{}f(a)=r</math>
The polynomial remainder theorem may be used to evaluate <math>f(a)</math> by calculating the remainder <math>r</math>. While polynomial long division is more difficult than evaluating the function itself, synthetic division is computationally easier. In the example in the polynomial long division article, the remainder of <math>f(x)/(x-3)</math> is 123 so <math>f(3)=123</math>.
The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear factor is a divisor. Repeated application of the factor theorem may be used to factorize the polynomial.it:Teorema del resto ja:剰余の定理 zh:多項式餘數定理