Remainder Theorem (Statement) MCQs Quiz | Class 9

This Class IX Mathematics (Code 041) quiz covers Unit II: Algebra, focusing specifically on the Remainder Theorem statement and finding the remainder on division by (x-a). Test your understanding of polynomial division concepts, submit your answers to see the score, and download the solution PDF for revision.

Understanding the Remainder Theorem

The Remainder Theorem is a fundamental concept in Class 9 Algebra that allows us to find the remainder when a polynomial is divided by a linear polynomial without performing the long division method. It connects the value of a polynomial at a specific point to the remainder obtained upon division.

Statement of the Theorem

Let p(x) be any polynomial of degree greater than or equal to 1, and let ‘a’ be any real number. If p(x) is divided by the linear polynomial (x – a), then the remainder is equal to p(a).

Key Concepts

  • Dividend: The polynomial p(x) being divided.
  • Divisor: The linear polynomial (x – a).
  • Remainder: The value p(a), which is a constant since the divisor is linear.
  • Degree Condition: The degree of the remainder is always less than the degree of the divisor. Since the divisor is linear (degree 1), the remainder must be a constant (degree 0).

How to Apply

To find the remainder when p(x) is divided by (x – a):

  1. Identify the value of ‘a’ from the divisor (x – a). If the divisor is (x + a), then use -a.
  2. Substitute this value into the polynomial p(x).
  3. Calculate the numerical value. This result is the remainder.
Polynomial p(x) Divisor Value to Substitute Remainder
3x^2 + x – 1 x – 1 x = 1 p(1) = 3(1)^2 + 1 – 1 = 3
x^2 – 4 x + 2 x = -2 p(-2) = (-2)^2 – 4 = 0

Revision Points

  • If p(a) = 0, then (x – a) is a factor of p(x).
  • The theorem only applies when the divisor is linear.
  • The remainder is independent of the quotient.

Extra Practice Questions

  1. Find the remainder when x^3 + 1 is divided by x + 1.
  2. Check if x – 2 is a factor of x^2 – 3x + 2 using the remainder theorem.
  3. If the remainder on dividing x^2 + kx + 3 by x – 1 is 5, find k.
  4. Divide 3x^4 – 4x^3 – 3x – 1 by x – 1 and verify the remainder.
  5. What is the remainder when the polynomial p(y) = y^3 + y^2 – 2y + 1 is divided by y + 3?