Chapter 2 — Polynomials

Chapter 2 of the Class 10 Maths NCERT textbook, "Polynomials", covers polynomials classified by degree, the geometrical meaning of their zeroes on a graph, and the relationships linking zeroes to the coefficients of quadratic and cubic polynomials.

• Classifying by degree — Polynomials are named by their highest power: degree 1 is linear, degree 2 quadratic, and degree 3 cubic. The degree also caps how many zeroes a polynomial can have — a quadratic at most two, a cubic at most three.
• Zeroes as graph crossings — A zero is a value where p(x) equals nothing, and geometrically it is where the curve y = p(x) meets the x-axis. A parabola may cross the axis at two, one, or no points, matching its number of real zeroes.
• Zeroes tied to coefficients — Without solving, you can predict the sum and product of a polynomial's zeroes directly from its coefficients. This link works for quadratics and extends to three symmetric relationships for cubic polynomials.

1. 01Polynomials of degree 1, 2, and 3 are called linear, quadratic, and cubic polynomials respectively
2. 02A real number k is a zero of polynomial p(x) if p(k) = 0; geometrically, zeroes are x-coordinates where the graph y = p(x) crosses the x-axis
3. 03A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes
4. 04For a quadratic ax² + bx + c with zeroes α and β: sum of zeroes = −b/a and product of zeroes = c/a
5. 05For a cubic ax³ + bx² + cx + d with zeroes α, β, γ: α+β+γ = −b/a, αβ+βγ+γα = c/a, and αβγ = −d/a
6. 06The graph of a quadratic polynomial is a parabola that can intersect the x-axis at 0, 1, or 2 points, corresponding to 0, 1, or 2 zeroes