Polynomials Exemplar — Class 9 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Polynomials Exemplar" — 6 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Polynomials Exemplar" — 6 important questions with detailed answers for CBSE board…
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NCERT Solutions for Class 9 Mathematics — Polynomials Exemplar. Step-by-step answers to all 6 textbook questions from this chapter, written for CBSE board preparation and free to use.
Polynomials Exemplar — All 6 Questions Solved
Q1. If p(x) = x^3 - 3x^2 + 2x, find p(-1) and p(2).
p(-1) = (-1)^3 - 3(-1)^2 + 2(-1) = -1 - 3 - 2 = -6. p(2) = (2)^3 - 3(2)^2 + 2(2) = 8 - 12 + 4 = 0.
Q2. Factorize: x^4 - 1.
x^4 - 1 = (x^2)^2 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1).
Q3. Divide x^4 + x^3 + x^2 + x + 1 by (x - 1) and find quotient and remainder.
Using synthetic division or long division: (x^4 + x^3 + x^2 + x + 1) ÷ (x - 1). At x = 1: p(1) = 1 + 1 + 1 + 1 + 1 = 5 (remainder). Quotient: x^3 + 2x^2 + 3x + 4 with remainder 5.
Q4. If x = 2 is a zero of p(x) = x^3 - 6x^2 + 11x - 6, factorize p(x).
Since x = 2 is a zero, (x - 2) is a factor. Dividing: p(x) = (x - 2)(x^2 - 4x + 3) = (x - 2)(x - 1)(x - 3).
Q5. Verify that x = -2 is a zero of p(x) = x^3 + 6x^2 + 11x + 6.
p(-2) = (-2)^3 + 6(-2)^2 + 11(-2) + 6 = -8 + 24 - 22 + 6 = 0. Yes, x = -2 is a zero.
Q6. Find all zeros of p(x) = x^3 - x^2 - x + 1.
p(x) = x^2(x - 1) - (x - 1) = (x - 1)(x^2 - 1) = (x - 1)(x - 1)(x + 1) = (x - 1)^2(x + 1). Zeros are x = 1 (double) and x = -1.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.