Chapter 2 of 14

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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 $- 4$ .

Worked Examples — Relationship Between Zeros and Coefficients

Example 1: Find a quadratic polynomial whose sum and product of zeros are −3 and 2 respectively. Formula: x² − (sum of zeros)x + (product of zeros) = x² − (−3)x + 2 = x² + 3x + 2. Verification: x²+3x+2 = (x+1)(x+2). Zeros: −1, −2. Sum = −3 ✓. Product = 2 ✓.

Example 2: If α and β are zeros of x² − 5x + k such that α−β = 1, find k. Sum: α+β = 5. Difference: α−β = 1. Solving: 2α = 6 → α = 3, β = 2. Product: αβ = k = 3×2 = 6.

Example 3 — Cubic Polynomial: Verify that 1, −2, and 3 are zeros of x³ − 2x² − 5x + 6. Check: 1³−2(1)²−5(1)+6 = 1−2−5+6 = 0 ✓. (−2)³−2(4)−5(−2)+6 = −8−8+10+6 = 0 ✓. 3³−2(9)−5(3)+6 = 27−18−15+6 = 0 ✓. Sum of zeros: 1+(−2)+3 = 2. Coefficient relationship: −(−2)/1 = 2 ✓. Product of zeros: 1×(−2)×3 = −6. Constant term with sign: −6/1 = −6 ✓.

Geometric Meaning of Zeros

The zeros of polynomial p(x) are the x-COORDINATES of the points where the graph y = p(x) intersects the X-AXIS. Linear polynomial: graph is a straight line — EXACTLY 1 zero. Quadratic polynomial: graph is a PARABOLA — at most 2 zeros. The parabola opens upward (∪) if the coefficient of x² > 0, downward (∩) if < 0. Cubic polynomial: S-curve — at most 3 zeros.

Division Algorithm for Polynomials

For any polynomials p(x) and g(x) (g(x) ≠ 0), there exist unique polynomials q(x) and r(x) such that: p(x) = g(x) × q(x) + r(x), where deg r(x) < deg g(x) or r(x) = 0.

Example: Divide x³ − 4x² + 5x − 2 by x−2. x³/x = x². Multiply: x²(x−2) = x³−2x². Subtract: −4x²−(−2x²) = −2x². Bring 5x: −2x²+5x. Divide: −2x²/x = −2x. Multiply: −2x(x−2) = −2x²+4x. Subtract: 5x−4x = x. Bring −2: x−2. Divide: x/x = 1. Multiply: 1(x−2) = x−2. Subtract: 0. Quotient: x²−2x+1 = (x−1)². Remainder: 0.

Key Identities (x+y+z Type)

(a+b+c)² = a²+b²+c²+2ab+2bc+2ca. a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca). If a+b+c = 0, then a³+b³+c³ = 3abc.

Common Mistakes

Confusing zeros and coefficients: The sum of zeros = −b/a, NOT b/a. The negative sign matters.
Graph of cubic ≠ two parabolas: A cubic has one S-curve shape that can cross the x-axis up to 3 times.
Division algorithm: The remainder's degree must be STRICTLY LESS than the divisor's degree.

CBSE Exam Focus

Topic	Marks
Geometric meaning of zeros	1-2
Relationship zeros ↔ coefficients	3-4
Division algorithm	3-4
Finding polynomial from zeros	2-3

Quick Self-Test

Find zeros of x² − 2x − 8. (Answer: 4 and −2.)
Sum and product of zeros of 3x² − 5x + 2? (Answer: Sum=5/3, Product=2/3.)
If α,β are zeros of x²−6x+k and α²+β²=20, find k. (Answer: α²+β²=(α+β)²−2αβ=36−2k=20 → k=8.)
Divide 2x³+x²−5x+2 by x+2. (Answer: Quotient=2x²−3x+1, Remainder=0. x+2 is a factor.)
Find a quadratic polynomial with zeros 3+√2 and 3−√2. (Answer: Sum=6, Product=9−2=7. Polynomial: x²−6x+7.)

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.

Q1MEDIUM

Find the zeros of 4x² − 4x − 3 and verify the relationship between coefficients and zeros.

Q2MEDIUM

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

Q3MEDIUM

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

Q4MEDIUM

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

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.
•To construct a quadratic with given zeros α, β: k[x² − (α+β)x + αβ] for any non-zero k.
•If remainder when p(x) divided by (x − a) is 0, then a is a zero of p(x) (Factor Theorem).

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: Polynomials sits in Unit II (Algebra), worth ~20 marks across the unit. This chapter contributes around 5 marks.

Question type	Marks each	Typical count	What it tests
MCQ / VSA (1 mark)	1	1	Identify degree, classify polynomial, or identify number of zeros from a graph
Short Answer (2 marks)	2	1	Find zeros + verify coefficient relations (2 marks), or construct polynomial from zeros
Long Answer (3 marks)	3	1	Polynomial division algorithm, or prove a relationship involving zeros

Prep strategy

The sum/product of zeros for quadratic (α + β = −b/a, αβ = c/a) must be instant recall. Write them 10 times without looking — they appear in EVERY polynomial question.
Division algorithm questions always ask for BOTH quotient and remainder — never leave one out. Also verify your answer by checking p(x) = g(x)·q(x) + r(x).
HOTS questions typically ask you to find an EXPRESSION involving the zeros without finding the zeros themselves. Practice (α + β)², (α − β)², (α + 1)(β + 1) type combinations.
Cubic polynomial zeros — memorise three formulas separately (sum, pairwise sum, triple product). The signs follow the pattern: −b/a, +c/a, −d/a.

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Signal processing — polynomials in disguise

Computer graphics — Bezier curves

Projectile motion — trajectory as a polynomial

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Sum/product formulas: write them on top of your answer sheet before starting any polynomial problem — this forces you to recall them at the start and prevents the sign error mistake.
Division algorithm verification step: always check your answer by multiplying back (g·q + r = p). This earns the verification mark AND catches computational errors — takes 30 seconds and is worth 1 mark.
HOTS questions: the strategy is NEVER to find zeros first. Always expand the expression using sum/product identities: (α+1)(β+1) = αβ + α + β + 1 = (product) + (sum) + 1. This algebraic manipulation is the skill being tested.
Graphs: the board may show a graph and ask for the number of zeros — count x-axis crossings/touches. A cubic can have 1, 2, or 3 crossings depending on the polynomial. Do not say 'quadratic always has 2 zeros.'

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Research Vieta's Formulas (François Viète, 16th century) — the general version of the sum/product of zeros relations for polynomials of any degree. For a degree-n polynomial, there are n Vieta relations connecting zeros to coefficients. These formulas are central to competition mathematics (IMO, USAMO) where finding sums of powers of roots, symmetric functions of roots, and polynomial constructions all use them.
Investigate the Fundamental Theorem of Algebra: every polynomial of degree n with complex coefficients has exactly n zeros in the complex numbers (counting multiplicity). Gauss proved this in 1799 for his doctoral thesis (the proof has been simplified many times since). What this means: a quadratic always has EXACTLY 2 complex zeros — they may be real and distinct, repeated real, or complex conjugate pairs. Which case applies depends on the discriminant b² − 4ac.
Polynomial roots and cryptography: the Reed-Solomon error correction code (used in QR codes, CDs, DVDs, and deep-space communication) is based on polynomial interpolation — given n points on a polynomial of degree n-1, you can uniquely reconstruct the polynomial. This is used to recover lost or corrupted data. The mathematical foundation is that a polynomial of degree n-1 is determined by n points (no more, no less).
Explore the concept of 'algebraic numbers' — numbers that are zeros of polynomials with rational coefficients. Are all real numbers algebraic? No — Cantor proved in 1874 that the algebraic numbers are countable but the reals are uncountable, so 'most' real numbers are TRANSCENDENTAL (not algebraic). Famous transcendental numbers: π and e. This connects the chapter's study of polynomial zeros to the deep structure of the real number line.

Where else this chapter is tested

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