Question

What is reminder theorem . please give answer with an example.

Answer 1

Remainder theorem:let p(x)be any polynomial of degree greater than or equal to one and let 'a'be any real number .If p(x)is divided by the linear polynomial (x-a),then the remainder is p(a).

ex:by division algorithm,
p(x)=g(x).q(x)+r(x)
p(x)=(x-a).q(x)+r(x)           g(x)=(x-a)
since the degree of(x-a)is 1 and the degree of r(x)is less than the degree of (x-a)
therefore,degree of r(x)=0,implies r(x)is a constant,say K
so,for every real value of x,r(x)=K
therefore,
p(x)=(x-a)q(x)+K
If x=a,then p(a)=(a-a)q(a)+K
                        =0+K
                        =K 
hence proved.

hope this helps u:))