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Schooling · Class 10 · Mathematics · Polynomials · Chapter on a page

CBSE Class 10 · Mathematics · NCERT Mathematics

Polynomials: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

This note's current text has no applied fact-check on record.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Polynomials: Key ideas, Formulas and facts, Watch out forKey ideasA polynomial in one variable x is an…The degree of a polynomial is the…A zero (or root) of a polynomial…A linear polynomial ax + b (a ≠ 0)…A quadratic polynomial ax² + bx + c…A cubic polynomial can have at most…The zeroes of a polynomial are found…Formulas and factsFor a quadratic polynomial ax² + bx…α + β = −b/aαβ = c/aFor a cubic polynomial ax³ + bx² +…α + β + γ = −b/aαβ + βγ + γα = c/aαβγ = −d/aWatch out forForgetting the negative sign in sum…Confusing zeroes with coefficientsAssuming every quadratic has two…Writing the division algorithm…Sign errors when the leading…Polynomials
The mind map as a list
  • Key ideas
    • A polynomial in one variable x is an…
    • The degree of a polynomial is the…
    • A zero (or root) of a polynomial…
    • A linear polynomial ax + b (a ≠ 0)…
    • A quadratic polynomial ax² + bx + c…
    • A cubic polynomial can have at most…
    • The zeroes of a polynomial are found…
  • Formulas and facts
    • For a quadratic polynomial ax² + bx…
    • α + β = −b/a
    • αβ = c/a
    • For a cubic polynomial ax³ + bx² +…
    • α + β + γ = −b/a
    • αβ + βγ + γα = c/a
    • αβγ = −d/a
  • Watch out for
    • Forgetting the negative sign in sum…
    • Confusing zeroes with coefficients
    • Assuming every quadratic has two…
    • Writing the division algorithm…
    • Sign errors when the leading…

What this chapter is about

This chapter extends your understanding of polynomials beyond what you studied in earlier classes. You learn to connect a polynomial's algebraic expression with its geometric picture — the graph. The focus is on how the zeroes of a polynomial relate to where its graph crosses the x-axis.

Key ideas

  • A polynomial in one variable x is an expression like 3x² − 5x + 2 where the powers of x are whole numbers (0, 1, 2, …) and coefficients are real numbers.
  • The degree of a polynomial is the highest power of the variable with a non-zero coefficient: degree 1 is linear, degree 2 is quadratic, degree 3 is cubic.
  • A zero (or root) of a polynomial p(x) is a value k such that p(k) = 0; geometrically, the graph of y = p(x) meets the x-axis at the point (k, 0).
  • A linear polynomial ax + b (a ≠ 0) has exactly one zero, namely −b/a.
  • A quadratic polynomial ax² + bx + c can have at most two zeroes; its graph is a parabola opening upward if a > 0 and downward if a < 0.
  • A cubic polynomial can have at most three zeroes; its graph is a smooth curve that can cross the x-axis up to three times.
  • The zeroes of a polynomial are found by factorising it or by using the relationship between zeroes and coefficients.
  • Division algorithm for polynomials: for polynomials p(x) and g(x) with g(x) ≠ 0, there exist polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

Formulas and facts to remember

  • For a quadratic polynomial ax² + bx + c with zeroes α and β:
  • Sum of zeroes: α + β = −b/a (negative of coefficient of x divided by coefficient of x²)
  • Product of zeroes: αβ = c/a (constant term divided by coefficient of x²)
  • For a cubic polynomial ax³ + bx² + cx + d with zeroes α, β and γ:
  • Sum of zeroes: α + β + γ = −b/a
  • Sum of products taken two at a time: αβ + βγ + γα = c/a
  • Product of all three zeroes: αβγ = −d/a
  • A polynomial of degree n has at most n zeroes.

Common mistakes

  • Forgetting the negative sign in sum of zeroes: students write α + β = b/a instead of −b/a → always put the minus sign first, then substitute values.
  • Confusing zeroes with coefficients: a zero is a value of x that makes the polynomial zero, not the numbers appearing in the expression → substitute the zero into the polynomial and check you get 0.
  • Assuming every quadratic has two distinct real zeroes: some quadratics touch the x-axis once (repeated zero) or not at all → the graph shows whether zeroes exist and how many.
  • Writing the division algorithm incorrectly: students forget that the remainder must have degree strictly less than the divisor → after dividing, check that the remainder's degree is lower.
  • Sign errors when the leading coefficient is negative: for −x² + 4x − 3, students misread a, b, c → rewrite clearly: a = −1, b = 4, c = −3, then apply formulas.

Quick revision

  • Degree tells the maximum number of zeroes a polynomial can have.
  • For ax² + bx + c: sum of zeroes = −b/a, product of zeroes = c/a.
  • Graph of a quadratic is a parabola; zeroes are where it crosses the x-axis.
  • Division algorithm: dividend = divisor × quotient + remainder.
  • Always verify zeroes by substituting back into the original polynomial.