Remainder Theorem


The remainder theorem, also known as Bezout’s Theorem is an application of Euclidean’s division of polynomials.

The remainder theorem definition says that when a polynomial f(x) is divided by the factor (x-a) and when the factor is not necessarily an element of the polynomial, then a smaller polynomial along with a reminder will also be found. The resulting obtained is the value of the polynomial f(x) where x = a and this is feasible only if f(a) = 0, in simple words in order to factorize polynomials, the remainder theorem is applied.

Remainder Theorem Proof

Consider a polynomial p(x) with a degree greater than or equal to 1, and, it is divided by another polynomial (x-k), where “k”’ is a real number.

And let q(x) and r(x) be quotient and remainder, we can write p(x) = (x-k)q(x) + r(x). The degree of the divisor (x-k) is 1. As, r(x) is the remainder, its degree is lesser than the degree of the divisor which is (x-k). Therefore, the degree of r(x) = 0, which says that r(x) is a constant.


p(x) = (x-k)q(x) + r

The value of p(x) at x = k is as follows-

p(k) = (k-k) q(k) + r

= (0)q(k) + r

= r

With this, we can infer that when a polynomial p(x) of a degree greater than or equal to one is divided by another linear polynomial (x-k), where “k” is any real number, then the remainder is r which is also equal to p(k).

Hence, the Reminder Theorem is proved.


Example 1: Check if p(x) = x4 + x3 – 2×2 + x + 1 is a multiple of (x-1).

Solution: Zero of divisor (x-1) is 1

Value of p(x) at x=1 is

p(x) = p(1) = 1+1-2+1+1= 2

This means when p(x) is divided by (x-1) we get 2 as the remainder.

Hence, p(x) is not a multiple of (x-1).


Example 2: Find the p(x) of the polynomial x4 – 2×3 + 4×2 – 5 if it is divided by (x-2).

Solution:Zero of divisor (x-2) is 2

Value of p(x) at x=2 is

p(x) = p(2) = 24 – 2(2)3 + 4(2)2 – 5 = 3

Points to Remember

  • Remainder Theorem says, “If p(x) is a polynomial of degree greater than or equal to one and is divided by the linear polynomial (x-k) where “k” is any real number, then the remainder is p(k).”
  • Remainder Theorem formula is given by the expression: p(x) = (x-k) q(x) + r(x).

Sample Questions

  1. State and Prove Remainder Theorem along with an example.
  2. Find the remainder when p(x) = x4 – x3 + x2 – 2x + 1 is divided by (x-2).
  3. Find the root of the polynomial x2 – 5x + 4
  4. Determine that x = 1 is a root of P(x).
  5. Find the remainder (without division) when 4×3 -3×2+2x-4 is divisible by (x+2).


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