Polynomials Exemplar — Class 10 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Polynomials Exemplar" — 5 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Polynomials Exemplar" — 5 important questions with detailed answers for CBSE board…
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NCERT Solutions for Class 10 Mathematics — Polynomials Exemplar. Step-by-step answers to all 5 textbook questions from this chapter, written for CBSE board preparation and free to use.
Polynomials Exemplar — All 5 Questions Solved
Q1. Find quotient and remainder when x^4 - 3x^2 + 4x + 5 is divided by x^2 - 1.
Using polynomial long division: x^4 - 3x^2 + 4x + 5 = (x^2 - 1)(x^2 - 2) + (4x + 7). Quotient = x^2 - 2, Remainder = 4x + 7.
Q2. If α and β are zeros of x^2 - 5x + 6, find α + β and αβ.
For ax^2 + bx + c: sum of zeros = -b/a, product = c/a. Here a = 1, b = -5, c = 6. So α + β = 5, αβ = 6. (Factoring: (x - 2)(x - 3), zeros are 2 and 3.)
Q3. Find a quadratic polynomial whose zeros are 3 and -7.
Sum = 3 + (-7) = -4, Product = 3 × (-7) = -21. Polynomial: x^2 - (sum)x + product = x^2 - (-4)x + (-21) = x^2 + 4x - 21.
Q4. Divide 4x^4 - 3x^3 - 6x + 8 by 2x^2 - 1.
Long division: 4x^4 - 3x^3 - 6x + 8 = (2x^2 - 1)(2x^2 - (3/2)x - (1/2)) + remainder. Quotient = 2x^2 - (3/2)x - (1/2), Remainder = -(15/2)x + (15/2).
Q5. If p(x) = x^3 - 6x^2 + 11x - 6 and one zero is 1, find all zeros.
Since p(1) = 1 - 6 + 11 - 6 = 0, (x - 1) is a factor. Dividing: p(x) = (x - 1)(x^2 - 5x + 6) = (x - 1)(x - 2)(x - 3). Zeros are 1, 2, 3.
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.