Remainder Theorem (Applications) MCQs Quiz | Class 9

This Class IX Mathematics (Code 041) quiz focuses on Unit II: Algebra, specifically the topic of Remainder Theorem (Applications). The questions cover key concepts such as finding the remainder quickly without long division and evaluating the polynomial P(a) for given values. Test your skills with these 10 multiple-choice questions, submit to view your detailed score, and download the solution PDF for future reference.

Understanding the Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra that helps us find the remainder when a polynomial is divided by a linear polynomial, without actually performing the long division process. This theorem is highly efficient for verifying factors and solving polynomial equations.

Key Statement

If a polynomial p(x) of degree greater than or equal to 1 is divided by a linear polynomial (x – a), then the remainder is simply p(a).

How to Apply

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

  1. Set the divisor equal to zero to find the value of x (the zero of the divisor).
  2. Substitute this value of x into the polynomial p(x).
  3. The calculated value is the remainder.

Quick Reference Table

Divisor Zero of Divisor Remainder
x – 2 x = 2 p(2)
x + 3 x = -3 p(-3)
2x – 1 x = 1/2 p(1/2)
ax + b x = -b/a p(-b/a)

Applications

  • Finding Remainders: Quickly determine the remainder for division problems.
  • Checking Factors: If the remainder is zero, then the divisor is a factor of the polynomial (Factor Theorem).
  • Solving for Unknowns: Calculate unknown coefficients (like k) when the remainder is given.

Practice Problems

Try these extra questions for practice:

  1. Find the remainder when x^3 – ax^2 + 6x – a is divided by x – a.
  2. Check if 7 + 3x is a factor of 3x^3 + 7x.
  3. If the polynomials ax^3 + 3x^2 – 13 and 2x^3 – 5x + a leave the same remainder when divided by x – 2, find the value of a.
  4. Find the remainder when x^51 + 51 is divided by x + 1.
  5. Use the Remainder Theorem to find the remainder when 4x^3 – 12x^2 + 14x – 3 is divided by 2x – 1.