    # Remainder Theorem Dividend = (Divisor � Quotient) + Remainder

If p(x) and t(x) are two polynomials such that degree of p(x) ≥ degree of t(x) and t(x) ≠ 0, then polynomials q(x) and r(x) can be found such that:

p(x) = t(x) q(x) + r(x),

Where :

r(x) = 0 or degree of r(x) < degree of t(x)

Or

It can be said that p(x) divided by t(x) gives q(x) as quotient and r(x) as remainder.

Statement of 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)."

Proof:

Let p(x) be any polynomial with degree of p(x) ≥ 1. Assuming that when p(x) is divided by x- a, the quotient is q(x) and the remainder is r(x), i.e. ,

p(x) = (x � a) q(x) + r(x)

Since the degree of x � a is 1 and the degree of r(x) is less than the degree of q(x), degree of r(x) = 0. This implies that r(a) is a constant, say r.

Hence for every value of x, r(x) = r.

Hence p(x) = (x �a)q(x) + r

If x = a, then the above equation will be:

p(a) = (a � a)q(a) + r

= r (hence proved)

Example:

Find the remainder when x3 + 2x2 + x + 1 is divided by x + 1.

Solution:

Let p(x) = x3 + 2x2 + x + 1

x + 1 = 0 x = -1

 Hence remainder = p(-1) = (-1)3 + 2(-1)2 + (-1) + 1 = -1 + 2 -1 + 1 = 1
Example:

Using long division method, divide the polynomial 3x4 � 4x3 � 3x - 1 by x � 1.

Solution:

 3x3 - x2 � x - 4 x � 1 3x4 � 4x3 � 3x � 1 3x4 � 3x3 -    + - x3 � 3x � 1 - x3 + x2 +    - -x2 - 3x -1 -x2 + x +    - - 4x � 1 - 4x + 4 +    - - 5

Hence Quotient = 3x3 - x2 � x � 4 and Remainder = -5.

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