Polynomials — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Polynomials (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Polynomials (CBSE & HBSE)
Covers zeros of polynomials and their geometric meaning. CBSE asks finding zeros from graphs and verifying relationships. HBSE tests the relationship between zeros and coefficients.
Zeros of Polynomials
Types of Polynomials
| Type | Degree | General Form | Example |
|---|---|---|---|
| Linear | 1 | ax + b | 2x + 3 |
| Quadratic | 2 | ax² + bx + c | x² - 5x + 6 |
| Cubic | 3 | ax³ + bx² + cx + d | x³ - 2x² + x - 1 |
Zero of a Polynomial
Definition: A value x = k is a zero of polynomial p(x) if p(k) = 0.
Geometric meaning:
- Zero = x-coordinate of the point where the graph of y = p(x) crosses the x-axis
- Linear polynomial → 1 zero (one x-intercept)
- Quadratic polynomial → at most 2 zeros (0, 1, or 2 x-intercepts)
- Cubic polynomial → at most 3 zeros
Number of Zeros from Graph
| Graph Shape | Cuts x-axis | Number of Zeros |
|---|---|---|
| Line | At 1 point | 1 zero |
| Upward parabola touching x-axis | 1 point | 1 zero (repeated) |
| Upward parabola cutting x-axis | 2 points | 2 zeros |
| Parabola above x-axis (no cut) | 0 points | No real zeros |
Finding Zeros
Linear: ax + b = 0 → x = -b/a Quadratic: by factorisation, completing square, or quadratic formula
Example: p(x) = x² - 3x - 10 Factorise: (x - 5)(x + 2) = 0 Zeros: x = 5 and x = -2
Exam Tip: To verify zeros, substitute back: p(5) = 25 - 15 - 10 = 0 ✓, p(-2) = 4 + 6 - 10 = 0 ✓
Relationship Between Zeros and Coefficients
For Quadratic Polynomial ax² + bx + c
If α and β are zeros:
| Relationship | Formula |
|---|---|
| Sum of zeros | α + β = −b/a |
| Product of zeros | α × β = c/a |
Memory trick: Sum = −(middle)/leading, Product = constant/leading
Forming a Quadratic from Given Zeros
If zeros are α and β: p(x) = x² − (α + β)x + αβ (or multiply by any non-zero constant k)
Example: Form quadratic with zeros 3 and −5.
- Sum = 3 + (−5) = −2 = −b/a
- Product = 3 × (−5) = −15 = c/a
- p(x) = x² − (−2)x + (−15) = x² + 2x − 15
For Cubic Polynomial ax³ + bx² + cx + d
If α, β, γ are zeros:
| Relationship | Formula |
|---|---|
| α + β + γ | −b/a |
| αβ + βγ + γα | c/a |
| α × β × γ | −d/a |
Verification Steps
- Compute zeros by factorisation
- Calculate sum and product of zeros
- Compare with −b/a and c/a respectively
- They should match!
CBSE Pattern: Often asks to verify relationship OR find a missing coefficient given one zero.
Division Algorithm for Polynomials
Division Algorithm
p(x) = g(x) × q(x) + r(x) Where:
- p(x) = Dividend
- g(x) = Divisor
- q(x) = Quotient
- r(x) = Remainder
- deg(r) < deg(g) OR r(x) = 0
Applications
- Verify whether g(x) divides p(x): Check if r(x) = 0
- Find remaining zeros: If one/two zeros known, divide and factor quotient
- Find quotient and remainder: Long division
Long Division Steps
- Arrange both polynomials in descending order of degree
- Divide leading term of dividend by leading term of divisor
- Multiply divisor by result and subtract
- Bring down next term, repeat
- Stop when degree of remainder < degree of divisor
Example: Divide p(x) = x³ - 3x² + 5x - 3 by g(x) = x² - 2
- x³ ÷ x² = x → multiply: x³ - 2x → subtract: -3x² + 7x - 3
- -3x² ÷ x² = -3 → multiply: -3x² + 6 → subtract: 7x - 9
- Quotient: x - 3, Remainder: 7x - 9
Verification: g × q + r = (x²-2)(x-3) + (7x-9) = x³-3x²-2x+6+7x-9 = x³-3x²+5x-3 ✓
HBSE Tip: Division algorithm questions often have partial information — one zero given, find all others using division.
Frequently asked questions
Are these Polynomials notes free?
Yes — the Polynomials notes for Mathematics (Class 10) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Polynomials notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Polynomials chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Polynomials.