CBSE • Class 9 • Mathematics • Chapter 2 (Polynomials) • Exercise 2.3

Exercise 2.3: Polynomials — NCERT Solutions

The Remainder Theorem — evaluating remainders without long division.

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Remainder theoremRemainder without divisionSubstitution trick

Step-by-step solutions — Exercise 2.3

2 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 2.3 Q1 • 2 marks

Using the Remainder Theorem, find the remainder when p(x)=x3−3x2+4x−5p(x) = x^3 - 3x^2 + 4x - 5 is divided by (x−2)(x - 2).
Hint (Socratic — try this first)▾
The Remainder Theorem says the remainder equals pp evaluated at which value?
Step-by-step solution▾

By the Remainder Theorem, the remainder when p(x)p(x) is divided by (x−a)(x - a) is p(a)p(a).

Here a=2a = 2, so compute p(2)p(2): p(2)=(2)3−3(2)2+4(2)−5=8−12+8−5=−1.p(2) = (2)^3 - 3(2)^2 + 4(2) - 5 = 8 - 12 + 8 - 5 = -1.

The remainder is −1-1.

Common mistake:
Substituting x=−2x = -2 instead of x=2x = 2; for divisor (x−a)(x-a) you use x=+ax = +a.
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Exercise 2.3 Q2 • 3 marks

Find the remainder when p(x)=4x3−12x2+14x−3p(x) = 4x^3 - 12x^2 + 14x - 3 is divided by (2x−1)(2x - 1).
Hint (Socratic — try this first)▾
Set the divisor equal to zero to find the value of xx you should substitute.
Step-by-step solution▾

First find the value of xx that makes the divisor zero: 2x−1=0  ⟹  x=12.2x - 1 = 0 \implies x = \tfrac12.

By the Remainder Theorem, remainder =p ⁣(12)= p\!\left(\tfrac12\right): p ⁣(12)=4(18)−12(14)+14(12)−3.p\!\left(\tfrac12\right) = 4\left(\tfrac18\right) - 12\left(\tfrac14\right) + 14\left(\tfrac12\right) - 3. =12−3+7−3=12+1=32.= \tfrac12 - 3 + 7 - 3 = \tfrac12 + 1 = \tfrac32.

The remainder is 32\dfrac{3}{2}.

Common mistake:
Using x=12x = \tfrac{1}{2} but mishandling the fractions, for example computing 4(12)34(\tfrac12)^3 as 42\tfrac{4}{2} instead of 48=12\tfrac{4}{8} = \tfrac12.
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How to approach Exercise 2.3

  1. Re-read the chapter summary first. Open Polynomials and refresh the key concepts: Polynomial, Degree, Remainder theorem, Factor theorem.
  2. Try each problem yourself before opening the solution. Ten focused minutes per problem usually beats reading two finished solutions.
  3. Use Guru AI for the questions you get stuck on. The Socratic AI tutor walks you through it with hints instead of dictating the answer.
  4. Mark the questions you got wrong and revisit them after 24 hours — the spacing is what locks the method into long-term memory.

All exercises in Polynomials

  1. Exercise 2.1 — Polynomials in one variable — degree, coefficients and zeros.
  2. Exercise 2.2 — Zeros of linear and quadratic polynomials — finding roots algebraically.
  3. Exercise 2.3 — The Remainder Theorem — evaluating remainders without long division.
  4. Exercise 2.4 — The Factor Theorem — detecting factors (x − a) from functional values.
  5. Exercise 2.5 — Algebraic identities — expanding and factorising cubic-type expressions.

← Exercise 2.2

Zeros of linear and quadratic polynomials — finding roots algebraically.

Exercise 2.4 →

The Factor Theorem — detecting factors (x − a) from functional values.

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