Class 10 Maths Notes — Chapter 2: Polynomials
Zeros of polynomials, their geometric meaning, and relationship with coefficients.
Detailed NCERT notes
- A polynomial in x is p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where aₙ ≠ 0. Degree = highest power of x.
- Types by degree: constant (0), linear (1), quadratic (2), cubic (3), biquadratic (4).
- A real number k is a zero of p(x) iff p(k) = 0. Geometrically, zero = x-coordinate of point where graph cuts x-axis.
- A polynomial of degree n has at most n real zeros.
- Quadratic ax² + bx + c: graph is a parabola. Opens upward if a > 0, downward if a < 0. Depending on discriminant D = b² − 4ac, parabola cuts x-axis at 2, 1 or 0 points.
- Relationship between zeros α, β of ax² + bx + c and coefficients: α + β = −b/a, αβ = c/a.
- Relationship for cubic ax³ + bx² + cx + d with zeros α, β, γ: α+β+γ = −b/a; αβ+βγ+γα = c/a; αβγ = −d/a.
- Given zeros, construct quadratic: x² − (sum)x + (product) = 0.
- Division Algorithm: for polynomials p(x) and g(x), g(x) ≠ 0, there exist unique q(x) and r(x) such that p(x) = g(x)·q(x) + r(x), with deg r(x) < deg g(x) or r(x) = 0.
- Long division of polynomials mirrors integer long division — arrange in decreasing degree, keep like terms aligned.
- If (x − a) is a factor of p(x), then p(a) = 0 (Factor Theorem — used implicitly).
- For a graph problem: count how many times the curve crosses the x-axis = number of real zeros.
Formulas & key results
- Quadratic: sum of zeros = −b/a, product = c/a
- Cubic: α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a
- Division: p(x) = g(x)·q(x) + r(x), deg r < deg g
- Quadratic from zeros: x² − (α+β)x + αβ
Mind map
- Zero of polynomial ↔ x-intercept of graph
- Quadratic graph = parabola
- Relations between zeros and coefficients (Vieta's)
- Division algorithm for polynomials
Tricks & shortcuts
- Verify zeros using Vieta's rather than resubstituting.
- For 'construct a polynomial with these zeros', straight to x² − Sx + P.
- Sketch parabola: sign of 'a' fixes opening; vertex at x = −b/2a.
Common mistakes to avoid
- Confusing 'zero of polynomial' with 'value of polynomial'.
- Sign errors in −b/a.
- Assuming a quadratic always has real zeros (check D).
Competency-based questions & answers
- Q. Find zeros of x² − 2x − 8 and verify the relations.A. (x − 4)(x + 2) = 0 ⇒ zeros 4, −2. Sum = 2 = −(−2)/1 ✓; Product = −8 = −8/1 ✓.
- Q. Form a quadratic whose zeros are 2 + √3 and 2 − √3.A. Sum = 4, Product = 4 − 3 = 1 ⇒ x² − 4x + 1 = 0.
- Q. If α, β are zeros of x² − 5x + 6, find α² + β².A. α+β = 5, αβ = 6; α² + β² = (α+β)² − 2αβ = 25 − 12 = 13.