Chapter 2 of 2

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CBSE Class 10 Mathematics★ 5 marks · CBSE Class 10 board

Polynomials

Zeroes of a polynomial, geometric meaning of zeroes, relationship between zeroes and coefficients, and the division algorithm for polynomials.

⏱ 10 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Identify the degree of a polynomial and classify it as linear, quadratic or cubic
2Read off the zeros of a polynomial from its graph
3Apply the relationship between coefficients and zeros for quadratic and cubic polynomials
4Use the polynomial division algorithm to divide one polynomial by another
5Construct a polynomial when its zeros are given

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Why this chapter matters

Polynomials are the language of algebra — quadratic equations, coordinate geometry, calculus and even programming algorithms are built on the idea of expressions in x. Master zeros and the division algorithm here and the next four chapters become 30% easier.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Real Numbers

Division algorithm idea + factorisation

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Linear Equations (Class 9)

Reading slope, intercepts from graphs

Polynomials

A polynomial in $x$ is an expression of the form $p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0},$ where the $a_{i}$ are real numbers, $a_{n} \neq = 0$ and $n$ is a non-negative integer. The degree is the highest power of $x$ that appears.

Degree	Name	Example
1	Linear	$2 x + 3$
2	Quadratic	$x^{2} - 5 x + 6$
3	Cubic	$x^{3} - x$

1. Zeroes of a Polynomial

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

Geometrically, the zeroes of $p (x)$ are the $x$ -coordinates where the graph $y = p (x)$ meets the $x$ -axis.

A linear polynomial has exactly one zero.
A quadratic has at most two zeroes.
A polynomial of degree $n$ has at most $n$ zeroes.

2. Relationship Between Zeroes and Coefficients

For a quadratic $p (x) = a x^{2} + b x + c$ with zeroes $α, β$ :

Sum of zeroes: $α + β = - \frac{b}{a}$ .
Product of zeroes: $α β = \frac{c}{a}$ .

For a cubic $p (x) = a x^{3} + b x^{2} + c x + d$ with zeroes $α, β, γ$ :

$α + β + γ = - \frac{b}{a}$
$α β + β γ + γ α = \frac{c}{a}$
$α β γ = - \frac{d}{a}$

Worked example — build a quadratic with given zeroes

Find a quadratic whose zeroes are $3$ and $- 2$ .

Sum $= 1$ , product $= - 6$ , so a polynomial is $p (x) = x^{2} - 1 \cdot x + (- 6) = x^{2} - x - 6.$

3. Division Algorithm for Polynomials

For polynomials $p (x)$ and $g (x)$ with $g (x) \neq = 0$ , there exist unique polynomials $q (x)$ and $r (x)$ such that $p (x) = g (x) \cdot q (x) + r (x),$ where $r (x) = 0$ or $de g r (x) < de g g (x)$ .

This is the polynomial analogue of integer division.

Worked example

Divide $p (x) = 3 x^{3} + x^{2} + 2 x + 5$ by $g (x) = x^{2} + 2 x + 1$ .

Long division gives: $p (x) = (x^{2} + 2 x + 1) (3 x - 5) + (9 x + 10) .$

Check: $de g (9 x + 10) = 1 < 2 = de g g (x)$ . ✓

Practice

Find the zeroes of $4 x^{2} - 4 x - 3$ and verify the sum/product relationships.
If $α, β$ are zeroes of $x^{2} - p (x + 1) - c$ , find $(α + 1) (β + 1)$ .
Divide $x^{3} - 3 x^{2} + 3 x - 5$ by $x - 1$ .

Answers

Zeroes are $\frac{3}{2}$ and $- \frac{1}{2}$ . Sum $= 1 = - (- 4) /4$ ✓. Product $= - \frac{3}{4} = - 3/4$ ✓.
Sum $= p$ , product $= - p - c$ . $(α + 1) (β + 1) = α β + α + β + 1 = (- p - c) + p + 1 = 1 - c$ .
Quotient $x^{2} - 2 x + 1$ , remainder $- 4$ .

Pre-requisites

You need Real Numbers — especially the division algorithm and the idea of unique factorisation — to follow §3 properly.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Quadratic zeros — sum

α + β = −b/a

For ax² + bx + c with zeros α, β.

Quadratic zeros — product

αβ = c/a

Same setup.

Cubic zeros — sum

α + β + γ = −b/a

For ax³ + bx² + cx + d.

Cubic zeros — pairwise

αβ + βγ + γα = c/a

Sum of products taken two at a time.

Cubic zeros — triple

αβγ = −d/a

Product of all three zeros.

Polynomial division

p(x) = g(x)·q(x) + r(x)

deg r(x) < deg g(x) or r(x) = 0.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Writing α + β = b/a (wrong sign) for ax² + bx + c

✓ Sum of zeros is −b/a. The negative sign comes from expanding (x−α)(x−β) = x² − (α+β)x + αβ.

WATCH OUT

✗ Saying a polynomial of degree 3 must have 3 real zeros

✓ It has at most 3 real zeros. It could have 1 real + 2 complex. The 'fundamental theorem of algebra' counts complex zeros with multiplicity.

WATCH OUT

✗ Skipping the degree check after polynomial division

✓ Always verify deg(r) < deg(g). If not, you haven't divided enough.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 2.1

Exercise 2.1

Reading zeros from a graph — visual interpretation

Questions

Ex 2.2

Exercise 2.2

Finding zeros and verifying the coefficient relationship

Questions

Ex 2.3

Exercise 2.3

Polynomial division algorithm

Questions

Ex 2.4

Exercise 2.4

Construct polynomials from given zeros (optional, but board-favourite)

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Coefficient relations

Find the zeros of 4x² − 4x − 3 and verify the sum/product of zeros.

▶ Show solution

Zeros: 3/2 and −1/2. Sum = 1 = −(−4)/4 ✓. Product = −3/4 = c/a = −3/4 ✓.

Q2MEDIUM· Construct from zeros

Find a quadratic polynomial with zeros 2 and −1/3.

▶ Show solution

Sum = 5/3, product = −2/3. Polynomial = k·(x² − (5/3)x − 2/3). Multiply by 3: 3x² − 5x − 2.

Q3MEDIUM· Division

Divide p(x) = x³ − 3x² + 3x − 5 by g(x) = x − 1.

▶ Show solution

Long division gives quotient x² − 2x + 1 and remainder −4. Check: (x−1)(x²−2x+1) + (−4) = x³ − 3x² + 3x − 5 ✓.

Q4HARD· HOTS

If α, β are zeros of x² − p(x+1) − c, find (α+1)(β+1).

▶ Show solution

Expand: x² − px − (p + c) so α + β = p and αβ = −(p+c). Then (α+1)(β+1) = αβ + α + β + 1 = −(p+c) + p + 1 = 1 − c.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Degree = highest power of x with non-zero coefficient.
•Zero of p(x) is a real k where p(k) = 0 — geometrically, where the graph cuts the x-axis.
•A polynomial of degree n has at most n real zeros.
•Quadratic ax² + bx + c with zeros α, β: α + β = −b/a, αβ = c/a.
•Cubic ax³ + bx² + cx + d with zeros α, β, γ: sum = −b/a, pair sum = c/a, product = −d/a.
•Division algorithm: p(x) = g(x)·q(x) + r(x) with deg(r) < deg(g) or r ≡ 0.

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Polynomials sits in Unit II (Algebra), worth ~20 marks across the unit. Polynomials itself contributes around 5 marks — typically 1 short-answer and 1 long-answer question, plus possibly an MCQ.

Exercise 2.4 (constructing polynomials from given zeros) is marked optional in NCERT, but board examiners frequently set questions from this exercise. Do it.

Quadratic Equations (Chapter 4) is just 'find the zeros of a quadratic polynomial' with extra techniques (factorisation, completing the square, quadratic formula). The sum/product results from this chapter recur there.

You won't be asked to plot from scratch. You will be asked to READ zeros from a given graph (Exercise 2.1 style) — make sure you can identify x-axis intersections.