Chapter 2 — Introduction to Polynomials | Class 9 CBSE Maths 2026-27

Section 2.1

Algebraic Expressions

Before studying polynomials, we need to understand what algebraic expressions are — the building blocks from which polynomials are made.

📦 What is an Algebraic Expression?

An algebraic expression is a combination of constants and variables connected by mathematical operations (+, −, ×, ÷). Variables represent unknown quantities, usually written as x, y, z, etc.

Anatomy of an algebraic expression: 3x² + 5x − 7 — each part has a specific name

Constant

7, −3

A fixed number with no variable. Its value never changes.

Variable

x, y, z

A letter that represents an unknown or changing quantity.

Coefficient

3 in 3x²

The number multiplied with a variable. Can be any real number.

Exponent

2 in x²

The power to which a variable is raised. Must be a whole number ≥ 0 in polynomials.

Term

3x², 5x, −7

Parts of an expression separated by + or −. Each coefficient-variable-exponent group.

Polynomial

p(x)

An algebraic expression where all exponents are non-negative whole numbers.

🔑 What makes something a POLYNOMIAL?

An expression is a polynomial in x if the exponents of x in every term are non-negative integers (0, 1, 2, 3, …)

✅ IS a polynomial: 3x² + 5x − 7 | x³ − 2 | 5 | x
❌ NOT a polynomial: 3x⁻¹ + 1 (negative exponent) | √x = x^(1/2) (fractional exponent) | 1/x

Section 2.2

Key Terminology

These are the precise mathematical names CBSE expects you to use. One wrong term can cost marks.

Term	Meaning	Example in p(x) = 4x³ − 2x² + 7x − 1
Polynomial in x	Expression with non-negative integer exponents of x	4x³ − 2x² + 7x − 1 ✓
Degree	Highest power of the variable	Degree = 3 (from x³)
Leading term	Term with the highest power	4x³
Leading coefficient	Coefficient of the leading term	4
Constant term	Term with no variable (power of x is 0)	−1
Number of terms	How many separate terms separated by + or −	4 terms
Coefficient of x²	Number multiplied with x²	−2
Coefficient of x	Number multiplied with x (= x¹)	7

💡

Exam Tip: When asked "find the coefficient of x²", always look for the number directly multiplying x². If the term is −2x², the coefficient is −2, NOT 2. The sign is part of the coefficient!

Section 2.3

Types of Polynomials

Polynomials are classified in two ways — by the number of terms, and by their degree.

Classification by Number of Terms

Monomial

1 term
Examples: 5, 3x, −7x², 4x³

3x²

Binomial

2 terms
Examples: x+1, 2x²−3, x³+x

x + 5

Trinomial

3 terms
Examples: x²+2x+1, 3x²−5x+2

x² + 2x + 1

Classification by Degree

Degree	Name	General Form	Example	Graph Shape
0	Constant	p(x) = a	p(x) = 5	Horizontal line
1	Linear	p(x) = ax + b	p(x) = 2x + 3	Straight line
2	Quadratic	p(x) = ax² + bx + c	p(x) = x² − 4	Parabola (U-shape)
3	Cubic	p(x) = ax³ + bx² + cx + d	p(x) = x³ − 3x	S-curve
n	nth degree	aₙxⁿ + ... + a₁x + a₀	x⁵ + 2x² − 1	Depends on n

Four main types of polynomials classified by degree

Section 2.4

Degree of a Polynomial

The degree is the highest exponent of the variable in the polynomial. Finding the degree correctly is one of the most tested skills in CBSE.

📐 Definition — Degree

Degree = the highest power of the variable in the polynomial

For p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ (where aₙ ≠ 0), the degree is n

Method

How to find the degree of p(x) = 5x³ − 4x² + 7x − 2

1

List all terms and their exponents:
5x³ → exponent 3 | −4x² → exponent 2 | 7x → exponent 1 | −2 → exponent 0

2

Find the highest exponent:
max(3, 2, 1, 0) = 3

3

State the degree:
Degree of p(x) = 3 Cubic polynomial

Degree = 3 (Cubic polynomial)

Polynomial	Terms	Highest Power	Degree	Type
7	1	x⁰ = 1	0	Constant
3x − 5	2	x¹	1	Linear
x² + 2x + 1	3	x²	2	Quadratic
2x³ − x + 4	3	x³	3	Cubic
x⁴ − 3x² + 2	3	x⁴	4	Bi-quadratic

⚠️

Tricky Cases: The degree of the zero polynomial (0) is undefined (or sometimes written as −∞). The degree of a non-zero constant like 5 is 0, because 5 = 5x⁰.

Section 2.5

Zeroes of a Polynomial

A zero (or root) of a polynomial p(x) is the value of x that makes p(x) = 0. Finding zeroes is a core skill tested in CBSE exams.

📌 Definition — Zero of a Polynomial

A real number a is called a zero of polynomial p(x) if p(a) = 0.
To find zeroes: set p(x) = 0 and solve for x.
A polynomial of degree n has at most n zeroes.

Example 1

Find the zero of p(x) = 2x − 6. Verify your answer. [2 Marks]

1

Set p(x) = 0:
2x − 6 = 0

2

Solve for x:
2x = 6 → x = 3

3

Verify — substitute x = 3 back into p(x):
p(3) = 2(3) − 6 = 6 − 6 = 0 ✓
💡 Verification is often asked for 1 extra mark in CBSE.

Zero of p(x) = 2x − 6 is x = 3

Example 2

If p(x) = x² − 3x + 2, find p(0), p(1), p(2). Which values are zeroes? [3 Marks]

1

Find p(0):
p(0) = (0)² − 3(0) + 2 = 0 − 0 + 2 = 2 → NOT a zero

2

Find p(1):
p(1) = (1)² − 3(1) + 2 = 1 − 3 + 2 = 0 ✓ → x = 1 IS a zero

3

Find p(2):
p(2) = (2)² − 3(2) + 2 = 4 − 6 + 2 = 0 ✓ → x = 2 IS a zero

4

Observation: x² − 3x + 2 = (x−1)(x−2), so it has exactly 2 zeroes. Degree 2 → at most 2 zeroes

Zeroes are x = 1 and x = 2. (p(0) = 2 is not a zero)

💡

Key Insight: Every linear polynomial ax + b (a ≠ 0) has exactly one zero: x = −b/a. This is because we solve ax + b = 0 → x = −b/a. There is always exactly one solution.

Section 2.6

Linear Polynomials: y = ax + b

A linear polynomial is a polynomial of degree 1. It has the form y = ax + b, where a is the slope and b is the y-intercept. Understanding these graphically is a key 2026-27 skill.

📐 Standard Form of a Linear Polynomial

y = ax + b (where a ≠ 0)

a = slope (how steep the line is; positive = rises left-to-right, negative = falls)
b = y-intercept (where the line crosses the y-axis)
Zero of this polynomial: x = −b/a (where the line crosses the x-axis)

📈 Slope (a)

📍 Y-intercept (b)

The y-intercept is where the line crosses the y-axis (where x = 0).

• b > 0: crosses above origin
• b < 0: crosses below origin
• b = 0: line passes through origin

Graphs of three linear polynomials — each is a straight line crossing the y-axis at its y-intercept

Example 3

For p(x) = 3x − 9, find (a) slope, (b) y-intercept, (c) zero, and (d) where line crosses axes. [3 Marks]

a

Slope: comparing with y = ax + b, a = 3 (line rises steeply upward)

b

y-intercept: b = −9 → the line crosses the y-axis at (0, −9)

c

Zero (x-intercept): set 3x − 9 = 0 → 3x = 9 → x = 3 → line crosses x-axis at (3, 0)

d

Axes crossing points: x-intercept = (3, 0), y-intercept = (0, −9)

Slope = 3, y-intercept = −9, Zero = x = 3, crosses at (3,0) and (0,−9)

Section 2.7

Reading Graphs of Polynomials

CBSE exam questions often show a graph and ask you to read off the zero, slope, or y-intercept. Here is how to do it systematically.

📊 How to read information from a graph of y = ax + b

1

Find the zero: The zero is the x-coordinate where the line crosses the x-axis (where y = 0). Just read the x-value at that point.

2

Find the y-intercept: Read the y-value where the line crosses the y-axis (where x = 0).

3

Find the slope: Pick two clear points on the line, say (x₁,y₁) and (x₂,y₂).
Slope a = (y₂ − y₁)/(x₂ − x₁)

4

Write the equation: Substitute a and b into y = ax + b.

🔑 Key Fact — Number of Zeroes from a Graph

• A linear polynomial (straight line) crosses the x-axis exactly once → 1 zero
• A quadratic polynomial (parabola) may cross the x-axis 0, 1, or 2 times
• A constant polynomial (horizontal line) crosses the x-axis 0 or infinitely many times
The number of times the graph crosses the x-axis = the number of real zeroes.

NCERT Style · CBSE Pattern

Worked Examples

These combine multiple polynomial concepts — the format mirrors 3–5 mark board questions.

Example 4

Find the degree, leading coefficient, and constant term of p(x) = 4 − 3x + 5x³ − x². Also write it in standard form. [3 Marks]

1

Rewrite in standard form (descending powers):
5x³ − x² − 3x + 4

2

Degree: Highest power = 3 (Cubic polynomial)

3

Leading coefficient: Coefficient of x³ = 5

4

Constant term: Term with no variable = 4

5

Coefficient of x²: = −1 (NOT 1 — the sign belongs to the coefficient)

Standard form: 5x³ − x² − 3x + 4 Degree = 3 | Leading coefficient = 5 | Constant term = 4

Example 5

Verify that x = 2 is a zero of p(x) = x³ − 3x² + 4. [2 Marks]

1

Substitute x = 2 into p(x):
p(2) = (2)³ − 3(2)² + 4

2

Calculate step by step:
= 8 − 3(4) + 4
= 8 − 12 + 4
= 0 ✓

3

Conclusion: Since p(2) = 0, x = 2 IS a zero of p(x). Verified

p(2) = 0 ✓ → x = 2 is a zero of p(x) = x³ − 3x² + 4

Example 6

If x = 3 is a zero of p(x) = kx² − 7x + 3, find k. [3 Marks]

1

Since x = 3 is a zero: p(3) = 0

2

Substitute x = 3:
k(3)² − 7(3) + 3 = 0
9k − 21 + 3 = 0

3

Solve for k:
9k − 18 = 0
9k = 18
k = 2

k = 2

NCERT Exercise

Exercise 2.1 — Fully Solved

All standard NCERT exercise questions with complete working — the format CBSE expects in answer papers.

Q1 · 1 Mark

Which of the following are polynomials?
(a) x² + 2x + 1 (b) x + 1/x (c) √x + 2 (d) x³ − 3

a

x² + 2x + 1: All exponents (2,1,0) are non-negative integers → Polynomial ✓

b

x + 1/x = x + x⁻¹: Exponent −1 is negative → NOT a polynomial ✗

c

√x + 2 = x^(1/2) + 2: Exponent 1/2 is a fraction → NOT a polynomial ✗

d

x³ − 3: Exponents (3,0) are non-negative integers → Polynomial ✓

Q2 · 2 Marks

Find the degree and coefficient of x² in: p(x) = 5x³ − 3x² + 2x − 1

1

Degree: highest power = 3 → Degree = 3

2

Coefficient of x²: the term with x² is −3x², so coefficient = −3

Degree = 3, Coefficient of x² = −3

Q3 · 2 Marks

Find the zero of p(x) = 3x + 9 and verify it.

1

Set p(x) = 0: 3x + 9 = 0

2

3x = −9 → x = −3

3

Verify: p(−3) = 3(−3) + 9 = −9 + 9 = 0 ✓

Zero = x = −3. Verified: p(−3) = 0

Q4 · 3 Marks

If p(x) = x² + x + k and p(2) = 0, find k. Then find the other zero.

1

p(2) = 0: (2)² + 2 + k = 0 → 4 + 2 + k = 0 → k = −6

2

So p(x) = x² + x − 6 = (x + 3)(x − 2)

3

Other zero: x + 3 = 0 → x = −3

k = −6. Zeroes are x = 2 and x = −3.

Smart Study

10 Study Tips for Polynomials

Targeted tips based on common CBSE mistakes.

1

Exponent must be a whole number ≥ 0

Fractions (½, ⅓) or negatives (−1, −2) as exponents disqualify it as a polynomial. x^(1/2) = √x is NOT a polynomial.

2

Zero polynomial ≠ zero of polynomial

The "zero polynomial" is the number 0 itself (degree undefined). A "zero of polynomial p(x)" is a value of x that makes p(x) = 0. These are completely different.

3

Coefficient includes the sign

In −5x², the coefficient is −5, not 5. Always include the negative sign with the coefficient. CBSE mark schemes check this carefully.

4

Constant has degree 0, not "no degree"

The polynomial 7 = 7×x⁰ has degree 0. Don't say "no degree" — say "degree 0". Only the zero polynomial has no degree.

5

Always write standard form for full marks

Standard form = descending order of powers. If question shows p(x) = 3 + x − 2x², rewrite as −2x² + x + 3 before answering.

6

Verify zeroes explicitly

When you find a zero, always substitute it back and show p(a) = 0. Verification is worth 1 mark in most CBSE 2–3 mark questions.

7

y = ax + b: slope and y-intercept

In y = 3x + 5, the slope is 3 (coefficient of x) and y-intercept is 5 (constant term). The zero is x = −5/3. These three are always connected.

8

Degree = max zeroes

A polynomial of degree n has at most n real zeroes. Linear → max 1 zero. Quadratic → max 2 zeros. Cubic → max 3 zeroes.

9

Missing terms have coefficient 0

In x³ + 5, the coefficient of x² is 0 and coefficient of x is 0. Don't say "there is no coefficient" — say the coefficient is 0.

10

Finding k from a zero — standard technique

If x = a is a zero and the polynomial contains k, substitute x = a, set equal to 0, then solve for k. This pattern appears in 2–3 mark CBSE questions every year.

Quick Reference

Chapter 2 — Formula & Fact Sheet

Concept	Rule / Formula	Notes
Polynomial in x	All exponents ∈ {0,1,2,3,...}	No fractions/negatives as powers
Degree	Highest exponent of variable	Constant polynomial has degree 0
Zero of p(x)	p(a) = 0 → a is a zero	Set p(x)=0 and solve
Zero of ax+b	x = −b/a	Linear polynomial has exactly 1 zero
Linear polynomial	y = ax + b (a ≠ 0)	Straight line graph
Slope	a in y = ax + b	Rate of change of y with x
Y-intercept	b in y = ax + b	Value of y when x = 0
Standard form	Descending powers of x	aₙxⁿ + ... + a₁x + a₀
Max zeroes	Degree n → at most n zeroes	Some may be complex/repeated

CBSE Pattern Practice

50 Practice Questions

MCQ · 1 Mark · 2 Marks · 3 Marks · 5 Marks · Case-Based — all with full step-by-step solutions.