Polynomials — Class 10 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Polynomials" — 9 important questions with detailed answers for CBSE board exam preparation.

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TL;DR: Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Polynomials" — 9 important questions with detailed answers for CBSE board exam pre…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

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Key Questions Covered:

State the relationship between zeroes and coefficients of a quadratic polynom…
Define a polynomial. What do you mean by the degree of a polynomial?
Divide p(x) = x³ + 3x² + 3x + 1 by g(x) = x + 1 using long division.
Find the zeroes of the polynomial p(x) = 2x² - 7x + 3.
If α and β are zeroes of x² - 4x + 3, find the polynomial whose zeroes are 2α…
Use the Remainder Theorem to find the remainder when p(x) = 2x³ - 3x² + 7x - …
+ 3 more questions in the full chapter

Solutions Summary:

Question	Status
State the relationship between zeroes and coefficients of…	✓ Solved
Define a polynomial. What do you mean by the degree of a …	✓ Solved
Divide p(x) = x³ + 3x² + 3x + 1 by g(x) = x + 1 using lon…	✓ Solved
Find the zeroes of the polynomial p(x) = 2x² - 7x + 3.	✓ Solved
If α and β are zeroes of x² - 4x + 3, find the polynomial…	✓ Solved
Use the Remainder Theorem to find the remainder when p(x)…	✓ Solved

Showing 6 of 9 questions

Q1: State the relationship between zeroes and coefficients of a quadratic polynomial ax² + bx + c.

For a quadratic polynomial ax² + bx + c with zeroes α and β: Sum of zeroes: α + β = -b/a Product of zeroes: α × β = c/a Example: For the polynomial 2x² - 5x + 3, with a = 2, b = -5, c = 3: If the zeroes are α and β, then: α + β = -(-5)/2 = 5/2 α × β = 3/2 We can verify: zeroes are 1 and 3/2 1 + 3/2 = 5/2 ✓ 1 × 3/2 = 3/2 ✓

Q2: Define a polynomial. What do you mean by the degree of a polynomial?

Polynomial: A polynomial is an algebraic expression consisting of terms with variables and coefficients, where variables have non-negative integer powers. General form: p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ where aₙ, aₙ₋₁, ..., a₁, a₀ are constants and n is a non-negative integer. Degree of a polynomial: The degree of a polynomial is the highest power of the variable in the polynomial. Examples: - p(x) = 3x² + 2x + 5 has degree 2 (quadratic) - p(x) = x³ - 4x + 1 has degree 3 (cubic) - p(x) ...

Q3: Divide p(x) = x³ + 3x² + 3x + 1 by g(x) = x + 1 using long division.

Polynomial long division of (x³ + 3x² + 3x + 1) ÷ (x + 1): Step 1: Divide first term of dividend by first term of divisor x³ ÷ x = x² Multiply: x²(x + 1) = x³ + x² Subtract: (x³ + 3x² + 3x + 1) - (x³ + x²) = 2x² + 3x + 1 Step 2: Repeat with new dividend 2x² + 3x + 1 2x² ÷ x = 2x Multiply: 2x(x + 1) = 2x² + 2x Subtract: (2x² + 3x + 1) - (2x² + 2x) = x + 1 Step 3: Repeat with new dividend x + 1 x ÷ x = 1 Multiply: 1(x + 1) = x + 1 Subtract: (x + 1) - (x + 1) = 0 Quotient: x² + 2x + 1 Remainder...

Q4: Find the zeroes of the polynomial p(x) = 2x² - 7x + 3.

Using the factorisation method: We need to find factors of 2 × 3 = 6 that add up to -7. The factors are -6 and -1 (since -6 + (-1) = -7 and -6 × (-1) = 6) 2x² - 7x + 3 = 2x² - 6x - x + 3 = 2x(x - 3) - 1(x - 3) = (2x - 1)(x - 3) Setting each factor to zero: 2x - 1 = 0 → x = 1/2 x - 3 = 0 → x = 3 Verification: For x = 1/2: 2(1/2)² - 7(1/2) + 3 = 2(1/4) - 7/2 + 3 = 1/2 - 7/2 + 3 = 0 ✓ For x = 3: 2(3)² - 7(3) + 3 = 18 - 21 + 3 = 0 ✓ Therefore, the zeroes are 1/2 a...

Q5: If α and β are zeroes of x² - 4x + 3, find the polynomial whose zeroes are 2α and 2β.

Step 1: Find the zeroes of x² - 4x + 3 Factorising: x² - 4x + 3 = (x - 1)(x - 3) Zeroes: α = 1, β = 3 Step 2: Find 2α and 2β 2α = 2(1) = 2 2β = 2(3) = 6 Step 3: Method 1 - Form polynomial with new zeroes Sum of new zeroes: 2α + 2β = 2 + 6 = 8 Product of new zeroes: (2α)(2β) = 2 × 6 = 12 Polynomial: x² - (sum)x + (product) = x² - 8x + 12 Step 4: Verification For x = 2: (2)² - 8(2) + 12 = 4 - 16 + 12 = 0 ✓ For x = 6: (6)² - 8(6) + 12 = 36 - 48 + 12 = 0 ✓ Therefore, the required polynomial is ...

Q6: Use the Remainder Theorem to find the remainder when p(x) = 2x³ - 3x² + 7x - 5 is divided by (x - 2).

Remainder Theorem: When a polynomial p(x) is divided by (x - a), the remainder is p(a). Here, p(x) = 2x³ - 3x² + 7x - 5 and we're dividing by (x - 2), so a = 2. Remainder = p(2) = 2(2)³ - 3(2)² + 7(2) - 5 = 2(8) - 3(4) + 14 - 5 = 16 - 12 + 14 - 5 = 13 Therefore, the remainder is 13

Showing 6 of 9 questions. Visit the full page for complete solutions.