Polynomials – Class 10 Maths Chapter 2 Notes & NCERT PDF

Class 10MathematicsChapter 2NCERT book: Mathematics - Class 10

Notes and a simple summary of Chapter 2 of the NCERT Class 10 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 2: Polynomials

Introduction

In Class IX you met polynomials in one variable and their degree. This chapter looks at the zeroes of a polynomial in two ways — geometrically (where the graph meets the x-axis) and algebraically (their relationship with the coefficients).

Key Concepts & Theorems

Polynomials of degree 1, 2 and 3 are called linear, quadratic and cubic polynomials.

A quadratic polynomial in x with real coefficients has the form ax² + bx + c, where a, b, c are real numbers and a ≠ 0.

A real number k is a zero of p(x) if p(k) = 0.

Geometrical meaning of zeroes

The zeroes of a polynomial p(x) are precisely the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.

A linear polynomial ax + b (a ≠ 0) has a straight-line graph meeting the x-axis at exactly one point, so it has exactly one zero, −b/a.

The graph of y = ax² + bx + c is a parabola, opening upwards if a > 0 and downwards if a < 0. It may cut the x-axis at two points, touch it at one point (two coincident zeroes) or not meet it at all (no real zero).

A quadratic has at most 2 zeroes; a cubic has at most 3. In general a polynomial of degree n has at most n zeroes (its graph meets the x-axis at most n times).

Example: the only zero of x³ is 0 — its graph meets the x-axis only at the origin.

Relationship between zeroes and coefficients

If α and β are the zeroes of ax² + bx + c:

α + β = −b/a and αβ = c/a

If α, β, γ are the zeroes of the cubic ax³ + bx² + cx + d:

α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a

A quadratic with zeroes whose sum is S and product is P is k(x² − Sx + P), for any non-zero constant k.

Methods Step by Step

Reading zeroes from a graph

Count the number of points where the curve meets the x-axis; that is the number of zeroes (touching counts as meeting at one point).

Finding zeroes and verifying the relationship

Factorise the quadratic (split the middle term or use an identity).

Set each factor to zero.

Compute sum and product of the zeroes and compare with −b/a and c/a.

Forming a quadratic from sum and product

Write x² − (sum)x + (product); multiply by a constant to clear fractions if needed.

Worked Examples

1. Zeroes of x² + 7x + 10. x² + 7x + 10 = (x + 2)(x + 5), so zeroes are −2 and −5. Sum = −7 = −(7)/1 ✓; product = 10 = 10/1 ✓.

2. Zeroes of x² − 3. Using a² − b² = (a − b)(a + b): (x − √3)(x + √3), so zeroes √3 and −√3. Sum = 0 = −0/1; product = −3 = −3/1.

3. A quadratic whose zeroes have sum −3 and product 2. Taking a = 1: b = 3, c = 2, giving x² + 3x + 2 (= (x + 1)(x + 2)).

4. Cubic check. For p(x) = 3x³ − 5x² − 11x − 3 the numbers 3, −1 and −1/3 are zeroes (substitute to get 0). Sum = 3 − 1 − 1/3 = 5/3 = −(−5)/3 ✓; sum of products in pairs = −3 + 1/3 − 1 = −11/3 = c/a ✓; product = 1 = −(−3)/3 ✓.

5. Exercise-style. Zeroes of 4x² − 4x − 3 = (2x − 3)(2x + 1) are 3/2 and −1/2; sum = 1 = 4/4, product = −3/4.

6. Forming from fractions. Sum 1/4, product −1: x² − (1/4)x − 1, or multiplying by 4, 4x² − x − 4.

Board Exam Focus

1 mark: "How many zeroes does the graph show?"; "Find the sum/product of zeroes of 2x² − 8x + 6" (sum 4, product 3).

2–3 marks: find zeroes of a quadratic and verify the relationship — show both the factorisation and the two checks.

Form a quadratic from given sum and product; accept any constant multiple, but write the standard form.

Formula Sheet

Linear ax + b: zero = −b/a

Quadratic: α + β = −b/a, αβ = c/a

Cubic: α + β + γ = −b/a; αβ + βγ + γα = c/a; αβγ = −d/a

Quadratic from S and P: k(x² − Sx + P)

Common Mistakes

Forgetting the minus sign in −b/a (sum) while the product c/a has none.

Counting the number of turns of a graph instead of its meetings with the x-axis.

Thinking every quadratic has two real zeroes — a parabola may not meet the x-axis at all.

Mixing the cubic product formula: it is −d/a, not d/a.

💡 Key Learning Points

✓Classify polynomials as linear, quadratic or cubic by degree, and know a quadratic has at most 2 and a cubic at most 3 zeroes.
✓Interpret zeroes geometrically as the x-coordinates where y = p(x) meets the x-axis.
✓Describe the parabola y = ax² + bx + c: opens up if a > 0, down if a < 0; may meet the x-axis at 2, 1 or 0 points.
✓Use α + β = −b/a and αβ = c/a for quadratics, and the three cubic relations.
✓Find zeroes by factorisation, verify the relationships, and form a quadratic from a given sum and product.

👨‍🏫 Teaching Tips

→Sketch several parabolas on a grid and ask students to count x-axis meetings before introducing the word 'zero'.
→Use graphing software or a table of values to show that changing the sign of a flips the parabola.
→Have students discover α + β and αβ from three factorised examples before stating the formulas.
→Drill 'sum = −b/a, product = c/a' with quick oral questions to fix the sign pattern.
→Pair work: one student gives a sum and product, the partner writes the quadratic; then swap.
→Show the cubic 3x³ − 5x² − 11x − 3 and verify all three relations as a class investigation.
→Link back to Chapter 4 by noting that zeroes of ax² + bx + c are the roots of ax² + bx + c = 0.

📋 Assessment Questions

Given a graph of y = p(x), can the student state the number of zeroes?
Can the student find the zeroes of x² − 2x − 8 and verify both relationships?
Can the student form a quadratic whose zeroes have sum 1/4 and product −1?
Can the student state the sign of a from the direction a parabola opens?
Can the student apply the cubic relations to verify given zeroes?

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