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Example:Based on Remainder Theorem

 Post to:   Explanation:

In mathematics, a polynomial (from Greek poly, "many" and medieval Latin binomium, "binomial") is an expression of finite length constructed from variables (also known as indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x2 − 4x + 7 is a polynomial, butx2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x (4/x) and because its third term contains an exponent that is not a whole number (3/2).

http://en.wikipedia.org/wiki/Polynomial

Polynomial remainder theorem

In algebra, the polynomial remainder theorem or little theorem is an application of polynomial long division. It states that the remainder of a polynomial divided by a linear divisor is equal to (Our solved example in mathguru.com uses this concept).

## Proof

The polynomial remainder theorem follows from the definition of polynomial long division; denoting the divisor, quotient and remainder by, respectively, , , and , polynomial long division gives a solution of the equation where the degree of is less than that of .

If we take as the divisor, giving the degree of as 0, i.e. : Setting we obtain:

Applications

The polynomial remainder theorem may be used to evaluate by calculating the remainder, r. Although polynomial long division is more difficult than evaluating the function itself, synthetic division is computationally easier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem.

The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial.

http://en.wikipedia.org/wiki/Polynomial_remainder_theorem