Polynomials Class 10 Notes (2026-27) — CBSE
Class 10 Maths Chapter 2 notes: zeroes of a polynomial, their geometrical meaning, and the relationship between zeroes and coefficients of quadratics and cubics.
Polynomials — Class 10 Maths Notes
Chapter Snapshot
This chapter is about the zeroes of a polynomial — what they mean, how they show up on a graph, and how they relate to the polynomial's coefficients. The star result is the sum/product of zeroes relationships for quadratic and cubic polynomials.
Board relevance: dependable Algebra scorer. Expect a "find the zeroes and verify the relationship with coefficients" question and a graph-reading question (count the zeroes from a figure).
Syllabus note (rationalised): the Division Algorithm for Polynomials section has been removed. Focus on zeroes and the zero–coefficient relationships.
Key Concepts & Definitions
Polynomial — an expression of the form aₙxⁿ + … + a₁x + a₀ with whole-number powers of x.
Degree — the highest power of x in the polynomial.
Degree Name General form
1 Linear ax + b
2 Quadratic ax² + bx + c
3 Cubic ax³ + bx² + cx + d
Zero of a polynomial — a value k for which p(k) = 0.
Geometrical meaning: the zeroes of p(x) are the x-coordinates of the points where the graph cuts the x-axis.
- A linear polynomial's graph is a straight line → at most 1 zero.
- A quadratic polynomial's graph is a parabola (∪ if a 0, ∩ if a < 0) → at most 2 zeroes (it may cut the x-axis at 2 points, touch it at 1, or miss it → 0 real zeroes).
- A polynomial of degree n has at most n zeroes.
Formulas
Quadratic polynomial ax² + bx + c (zeroes α, β)
Sum of zeroes: α + β = −b/a
Product of zeroes: αβ = c/a
Cubic polynomial ax³ + bx² + cx + d (zeroes α, β, γ)
α + β + γ = −b/a
αβ + βγ + γα = c/a
αβγ = −d/a
Forming a quadratic from its zeroes
p(x) = k[ x² − (α + β)x + αβ ] (take k = 1 for the simplest form)
Worked Examples
Example 1 — Find zeroes and verify: Find the zeroes of x² − 2x − 8 and verify the relationships.
Factorise: x² − 2x − 8 = (x − 4)(x + 2) → zeroes 4 and −2.
Check: sum = 4 + (−2) = 2 = −b/a = −(−2)/1 ✓; product = 4 × (−2) = −8 = c/a = −8/1 ✓.
Example 2 — Form a polynomial: Find a quadratic whose zeroes are 3 and −2.
Sum = 1, product = −6 → x² − (1)x + (−6) = x² − x − 6.
Example 3 — Use the relationship: If α, β are the zeroes of x² − 5x + 6, find 1/α + 1/β.
Sum = 5, product = 6. 1/α + 1/β = (α + β)/(αβ) = 5/6 = 5/6.
Example 4 — Cubic: Verify the relationships for 2x³ − 5x² − 14x + 8 given a zero pattern.
For a cubic, α + β + γ = −b/a = 5/2; αβ + βγ + γα = c/a = −14/2 = −7; αβγ = −d/a = −8/2 = −4. (Use these to check computed zeroes.)
Example 5 — Find α² + β²: If α and β are the zeroes of x² − 6x + 8, find α² + β².
Sum = 6, product = 8. Use α² + β² = (α + β)² − 2αβ = 6² − 2(8) = 36 − 16 = 20. (This "find a symmetric expression without finding the zeroes" trick is extremely common — always reach for the sum and product first.)
Example 6 — Find the polynomial from a condition: Find a quadratic polynomial the sum of whose zeroes is −3 and the product is 2.
p(x) = x² − (sum)x + product = x² − (−3)x + 2 = x² + 3x + 2. Check by factorising: (x + 1)(x + 2), zeroes −1 and −2, whose sum is −3 and product 2 ✓.
Notice how, in every example, the two relationships α + β = −b/a and αβ = c/a do all the work — you rarely need to find the zeroes explicitly.
Important Question Patterns
1. Find zeroes & verify (3 marks): factorise the quadratic, state both zeroes, then verify sum = −b/a and product = c/a.
2. Form a polynomial (2 marks): from given zeroes using x² − (sum)x + product; or from a given sum and product.
3. Use the relationships (2–3 marks): find expressions like 1/α + 1/β, α² + β² = (α+β)² − 2αβ, or α − β without finding the zeroes explicitly.
4. Graph reading (1–2 marks): count the number of zeroes from a graph (how many times it crosses the x-axis); identify the sign of the leading coefficient.
5. Cubic relationships (2–3 marks): verify the three cubic relations, or find the third zero given two.
⚡ Quick Revision
- Zero: p(k) = 0; geometrically, where the graph cuts the x-axis.
- Degree n → at most n zeroes. Quadratic graph = parabola (∪ or ∩); may have 2, 1, or 0 real zeroes.
- Quadratic (α, β): sum α + β = −b/a, product αβ = c/a.
- Cubic (α, β, γ): α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a.
- Form a quadratic: x² − (sum)x + product.
- Handy identity: α² + β² = (α + β)² − 2αβ.
- Removed from syllabus: division algorithm for polynomials.
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