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Example:Based on Remainder Theorem

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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

 

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.