Remainder Theorem

Remainder Theorem: If a polynomial f ( x ) is divided by ( x - a ), then the remainder = f ( a )

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 a linear polynomial x - a , then the remainder is p ( a ).

Proof for Remainder theorem in Algebra

Proof: Suppose when p ( x ) is divided by x - a , the quotient is q ( x ) and the remainder is r ( x ).
Then p ( x ) = ( x - a ) q ( x ) + r ( x )
where degree r ( x ) <>
Since, degree ( x -a ) = 1, therefore r ( x ) is a constant , say r
For all values of x,
p ( x ) = ( x - a ) q ( x ) + r
for x = a ,
p ( a ) = ( a - a ) q ( a ) + r
= 0 x q ( a ) + r
= r
Hence the theorem is proved.