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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Key Questions Covered:
- Find quotient and remainder when x^4 - 3x^2 + 4x + 5 is divided by x^2 - 1.
- If α and β are zeros of x^2 - 5x + 6, find α + β and αβ.
- Find a quadratic polynomial whose zeros are 3 and -7.
- Divide 4x^4 - 3x^3 - 6x + 8 by 2x^2 - 1.
- If p(x) = x^3 - 6x^2 + 11x - 6 and one zero is 1, find all zeros.
Solutions Summary:
| Question | Status |
|---|---|
| Find quotient and remainder when x^4 - 3x^2 + 4x + 5 is d… | ✓ Solved |
| If α and β are zeros of x^2 - 5x + 6, find α + β and αβ. | ✓ Solved |
| Find a quadratic polynomial whose zeros are 3 and -7. | ✓ Solved |
| Divide 4x^4 - 3x^3 - 6x + 8 by 2x^2 - 1. | ✓ Solved |
| If p(x) = x^3 - 6x^2 + 11x - 6 and one zero is 1, find al… | ✓ Solved |
Showing 5 of 5 questions
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.
Showing 5 of 5 questions. Visit the full page for complete solutions.
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