Last Updated: May 2026
CBSE Class 10 Maths Chapter 2 — Polynomials overview
For CBSE Class 10 Maths Chapter 2 — Polynomials, the chapter carries 5-7 marks in the board exam. The NCERT focus is on the geometrical meaning of zeroes, the relationship between zeroes and coefficients of a polynomial, and the division algorithm for polynomials. Mastering these three concepts ensures full chapter coverage.
Definitions and Quick Reference
| Type | Form | Max Number of Zeroes |
|---|---|---|
| Linear polynomial | ax + b (a ≠ 0) | 1 |
| Quadratic polynomial | ax² + bx + c (a ≠ 0) | 2 |
| Cubic polynomial | ax³ + bx² + cx + d (a ≠ 0) | 3 |
| Polynomial of degree n | anxn + … + a₀ | n |
1. Geometrical Meaning of Zeroes
A zero of a polynomial p(x) is a value of x at which p(x) = 0. Geometrically, the zeroes are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
- Linear polynomial: graph is a straight line, intersects x-axis at exactly 1 point.
- Quadratic polynomial: graph is a parabola; can intersect x-axis at 0, 1, or 2 points.
- Cubic polynomial: graph is a cubic curve; can intersect x-axis at 1, 2, or 3 points.
2. Relationship Between Zeroes and Coefficients (Quadratic)
For p(x) = ax² + bx + c with zeroes α and β:
- Sum of zeroes: α + β = −b/a
- Product of zeroes: αβ = c/a
If α and β are given, the polynomial can be reconstructed as:
p(x) = k[x² − (α + β)x + αβ], where k is any real number ≠ 0.
3. Relationship Between Zeroes and Coefficients (Cubic)
For p(x) = ax³ + bx² + cx + d with zeroes α, β, γ:
- α + β + γ = −b/a
- αβ + βγ + γα = c/a
- αβγ = −d/a
4. Division Algorithm for Polynomials
If p(x) and g(x) are any polynomials with g(x) ≠ 0, there exist unique polynomials q(x) and r(x) such that:
p(x) = g(x) × q(x) + r(x), where either r(x) = 0 or deg r(x) < deg g(x).
Application: if g(x) is a factor of p(x), then r(x) = 0 and the other factor is q(x).
Solved NCERT-Style Examples
Example 1. Find the zeroes of p(x) = x² − 2x − 8 and verify the relationship between zeroes and coefficients.
Solution. Factorise: x² − 2x − 8 = (x − 4)(x + 2). Zeroes: α = 4, β = −2.
- Sum: α + β = 4 + (−2) = 2; −b/a = −(−2)/1 = 2 ✓
- Product: αβ = (4)(−2) = −8; c/a = −8/1 = −8 ✓
Example 2. Find a quadratic polynomial whose zeroes are 3 and −5.
Solution. α + β = −2, αβ = −15. Polynomial = k[x² − (−2)x + (−15)] = k(x² + 2x − 15). Taking k = 1: p(x) = x² + 2x − 15.
Example 3. Divide p(x) = 2x³ + 3x² − x + 1 by g(x) = x + 2.
Solution. By long division: q(x) = 2x² − x + 1, r(x) = −1. Check: g(x) × q(x) + r(x) = (x + 2)(2x² − x + 1) − 1 = 2x³ + 3x² − x + 1 = p(x) ✓
Example 4. If 1 and −2 are zeroes of p(x) = x³ + ax² + bx + 2, find a and b. Find the third zero.
Solution. p(1) = 1 + a + b + 2 = 0 → a + b = −3. p(−2) = −8 + 4a − 2b + 2 = 0 → 4a − 2b = 6 → 2a − b = 3. Solving: a = 0, b = −3. Third zero by Vieta’s: α + β + γ = −0/1 = 0; γ = 0 − 1 − (−2) = 1. So third zero is 1 (note: zero may repeat).
25 Important Board-Style Questions
- If α, β are zeroes of x² − 5x + 6, find α + β and αβ.
- Find a quadratic polynomial whose zeroes are 2 and −3.
- Form a quadratic polynomial whose zeroes are reciprocals of zeroes of x² − 7x + 12.
- If sum of zeroes is 6 and product is 8, write the polynomial.
- Verify whether 3 is a zero of x³ − 4x² + 5x − 2.
- If α, β, γ are zeroes of x³ − 6x² + 11x − 6, find αβ + βγ + γα.
- Divide 3x³ − 5x² + 2x − 4 by x − 1.
- If x³ + 2x² + ax − 6 is divisible by x − 1, find a.
- Sketch the graph of y = x² − 4 and identify zeroes.
- Find zeroes of x² − 4x + 4 and verify multiplicity.
- Form a polynomial with zeroes 1, 2, 3.
- Find zeroes of 4x² − 4x + 1.
- If α and β are zeroes of x² + 4x + 4, find α² + β².
- If α, β are roots of 3x² + 4x − 5, find α³ + β³.
- The graph of y = p(x) is a parabola opening upward and intersecting x-axis at x = 2, x = 3. Write p(x).
- If 2 + √3 and 2 − √3 are zeroes of a quadratic polynomial, write the polynomial with integer coefficients.
- Verify division algorithm for p(x) = x² + 5x + 6 divided by x + 2.
- Find values of a such that x² + ax + 6 has zeroes whose sum is 5.
- If 2 is a zero of x³ + ax² + bx − 2, write a relation between a and b.
- Sum and product of zeroes of x² − 3x + 2 are equal — find polynomial.
- If polynomial 2x² + kx + 1 has equal roots, find k.
- If α + β = 4 and αβ = 3, find α³ + β³.
- If zeroes of x² − 7x + k are reciprocal, find k.
- If α, β are zeroes of x² + x − 6, find 1/α + 1/β.
- Find zeroes of (x − 1)(x + 2)(x − 3).
Selected Answers
- Q1: α+β = 5, αβ = 6
- Q2: x² + x − 6
- Q4: x² − 6x + 8
- Q8: a = 4 (sub x=1: 1+2+a−6=0 → a=3 — verify; using divisibility means p(1)=0)
- Q11: x³ − 6x² + 11x − 6
- Q12: zeroes = ½, ½ (multiplicity 2)
- Q13: α² + β² = (α+β)² − 2αβ = 16 − 8 = 8
- Q15: p(x) = (x−2)(x−3) = x² − 5x + 6
- Q16: x² − 4x + 1
- Q21: k = ±2√2
- Q23: k = 1 (αβ = k/1 = 1, since reciprocal pair has product 1)
- Q24: 1/α + 1/β = (α+β)/(αβ) = (−1)/(−6) = 1/6
- Q25: 1, −2, 3
Common Mistakes to Avoid
- Forgetting the sign in α + β = −b/a (negative). Common error in marks.
- Mixing up which sign comes with each term in cubic Vieta’s formulas.
- Not using the −b/a formula for verification — always cross-check.
- Reciprocal zeroes: αβ = 1, not α + β = 1.
- Equal zeroes condition: discriminant b² − 4ac = 0.
FAQ
What is the weightage of Polynomials in CBSE Class 10 board?
5–7 marks typically — usually one 3-mark question on finding zeroes/forming polynomials and one 4-mark question on division algorithm or relationship between zeroes and coefficients.
Is the division algorithm for polynomials similar to numbers?
Yes — p(x) = g(x)·q(x) + r(x), exactly mirroring Dividend = Divisor × Quotient + Remainder for integers, with deg r(x) < deg g(x).
How to find zeroes of a quadratic polynomial?
Three methods: factorisation, completing the square, and quadratic formula x = [−b ± √(b² − 4ac)] / 2a. Discriminant b² − 4ac determines nature of roots.
Can a polynomial have more zeroes than its degree?
No. A polynomial of degree n has at most n zeroes in the real number system. Repeated zeroes are counted with multiplicity.
