CBSE • Class 9 • Mathematics • Chapter 2 (Polynomials) • Exercise 2.4

Exercise 2.4: Polynomials — NCERT Solutions

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

Aligned to the latest NCERT 2024-25 edition • 5 questions in this exercise • Free plan, no credit card

Open this exercise in the AI tutorBack to Chapter 2

What this exercise covers

Factor theoremTesting factorsFactorisation

Step-by-step solutions — Exercise 2.4

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

Exercise 2.4 Q1 • 2 marks

Use the Factor Theorem to determine whether (x+1)(x + 1) is a factor of p(x)=x3+3x2+3x+1p(x) = x^3 + 3x^2 + 3x + 1.
Hint (Socratic — try this first)▾
What must p(−1)p(-1) equal for (x+1)(x + 1) to be a factor?
Step-by-step solution▾

By the Factor Theorem, (x+1)=(x−(−1))(x + 1) = (x - (-1)) is a factor of p(x)p(x) if and only if p(−1)=0p(-1) = 0.

p(−1)=(−1)3+3(−1)2+3(−1)+1=−1+3−3+1=0.p(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1 = -1 + 3 - 3 + 1 = 0.

Since p(−1)=0p(-1) = 0, (x+1)(x + 1) is a factor of p(x)p(x).

Common mistake:
Substituting x=1x = 1 for the factor (x+1)(x+1); the correct value to test is x=−1x = -1.
Open this question in the AI tutor →

Exercise 2.4 Q2 • 3 marks

Find the value of kk if (x−2)(x - 2) is a factor of p(x)=2x3−3x2+kx−2p(x) = 2x^3 - 3x^2 + kx - 2.
Hint (Socratic — try this first)▾
Which equation involving kk do you get by applying the Factor Theorem?
Step-by-step solution▾

If (x−2)(x - 2) is a factor, then by the Factor Theorem p(2)=0p(2) = 0.

p(2)=2(2)3−3(2)2+k(2)−2=16−12+2k−2=2+2k.p(2) = 2(2)^3 - 3(2)^2 + k(2) - 2 = 16 - 12 + 2k - 2 = 2 + 2k.

Set equal to zero: 2+2k=0  ⟹  2k=−2  ⟹  k=−1.2 + 2k = 0 \implies 2k = -2 \implies k = -1.

So k=−1k = -1.

Common mistake:
Forgetting to set the expression equal to 00, or making a sign error while collecting constant terms.
Open this question in the AI tutor →

How to approach Exercise 2.4

  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 2.3

The Remainder Theorem — evaluating remainders without long division.

Exercise 2.5 →

Algebraic identities — expanding and factorising cubic-type expressions.

Solve Exercise 2.4 with AI guidance

Free plan. No credit card. Works on any device.

Start Free