Chapter 2: Polynomials | SJMaths Class 9

1. Basics of Polynomials

A polynomial is an algebraic expression where the exponent of the variable is always a whole number.

Classification by Terms

Terms	Name	Example
1 Term	Monomial	\( 2x, 5, 3x^2 \)
2 Terms	Binomial	\( x + 1, x^2 - 4 \)
3 Terms	Trinomial	\( x^2 + x + 1 \)

Polynomial Checker

Is \( x + \frac{1}{x} \) a polynomial?

1.1 Understanding the Building Blocks

Imagine this: A box 📦 whose value keeps changing.

Variable: The letter whose value can change (like x)
Constant: A fixed number (like 5)
Coefficient: The number multiplied with a variable

Example: In \( 3x^2 + 5x - 7 \)

Variable → x
Coefficient of \( x^2 \) → 3
Constant term → -7

🧠 Think & Guess

In \( -4x \), what is the coefficient?

Show Answer

-4

2. Degree of a Polynomial

The highest power of the variable is the degree.

Degree	Name	Standard Form
Not Defined	Zero Polynomial	\( 0 \)
0	Constant Polynomial	\( 7, -5, \frac{3}{2} \)
1	Linear	\( ax + b \)
2	Quadratic	\( ax^2 + bx + c \)
3	Cubic	\( ax^3 + bx^2 + cx + d \)

2.1 Degree vs Zeroes

Important CBSE Result:

Degree 1 → At most 1 zero
Degree 2 → At most 2 zeroes
Degree 3 → At most 3 zeroes

The graph may have fewer zeroes, but never more than its degree.

Why is this true?

Higher degree means more bending ability of the graph, allowing more x-axis crossings.

Quick Check

What is the degree of: \( 4x^5 - 2x^7 + 1 \)

5

7

0

1.2 Polynomial or Not? (Exam Traps 🚨)

A polynomial must satisfy:

No variable in denominator
No negative powers
No fractional powers

Expression	Polynomial?	Reason
\( 3x^2 - 5x + 1 \)	Yes	All powers are whole numbers
\( \frac{1}{x} + 2 \)	No	Power of x is -1
\( \sqrt{x} + 1 \)	No	Power is 1/2

⚠️ CBSE TRAP: Any fractional or negative power → NOT a polynomial

3. Remainder & Factor Theorems

Division Algorithm:
Dividend = (Divisor × Quotient) + Remainder

Remainder Theorem

If a polynomial \( p(x) \) is divided by \( x - a \), the remainder is simply \( p(a) \).

Example: Find remainder when \( p(x) = x^3 - 1 \) is divided by \( x - 1 \).

\( p(1) = (1)^3 - 1 = 0 \). Remainder is 0.

Factor Theorem

1. If \( p(a) = 0 \), then \( (x - a) \) is a factor of \( p(x) \).

2. If \( (x - a) \) is a factor of \( p(x) \), then \( p(a) = 0 \).

3.1 Zeroes of a Polynomial (Graph View)

Imagine this: A road (x-axis) and a curve crossing it.

A zero of a polynomial is the value of x where the graph cuts the x-axis.

Key Rule:

Graph touches x-axis → one zero
Graph crosses x-axis twice → two zeroes
Graph never touches x-axis → no real zero

🧠 CBSE Thinking

CBSE uses graphs to test whether students understand zeroes visually, not just algebraically.

⚠️ Zeroes ≠ solutions always. They depend on graph intersection.

4. Factorization Methods

Method 1: Splitting the Middle Term (Quadratics)

For \( ax^2 + bx + c \), we find two numbers \( p, q \) such that:

Sum \( p + q = b \)
Product \( pq = a \times c \)

Example: \( x^2 - 5x + 6 \)
Sum = -5, Product = 6. Numbers are -2 and -3.
\( x^2 - 2x - 3x + 6 \Rightarrow x(x-2) - 3(x-2) \Rightarrow (x-2)(x-3) \)

Method 2: Using Factor Theorem (Cubics)

For cubic polynomials, first find one factor by trial (usually \(\pm 1, \pm 2\)), then divide the polynomial by that factor to get a quadratic.

5. Algebraic Identities (The Big 8)

These are the standard identities for Class 9.

I. \( (x + y)^2 \)

\( x^2 + 2xy + y^2 \)

II. \( (x - y)^2 \)

\( x^2 - 2xy + y^2 \)

III. \( x^2 - y^2 \)

\( (x + y)(x - y) \)

IV. \( (x + a)(x + b) \)

\( x^2 + (a + b)x + ab \)

V. \( (x + y + z)^2 \)

\( x^2 + y^2 + z^2 + 2xy + 2yz + 2zx \)

VI. \( (x + y)^3 \)

\( x^3 + y^3 + 3xy(x + y) \)

VII. \( (x - y)^3 \)

\( x^3 - y^3 - 3xy(x - y) \)

VIII. \( x^3 + y^3 + z^3 - 3xyz \)

\( (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \)

Special Case: If \( x + y + z = 0 \), then \( x^3 + y^3 + z^3 = 3xyz \).

🧠 How to Choose the Correct Identity?

Look for perfect squares → use \( (a+b)^2 \)
Look for subtraction → use \( a^2-b^2 \)
Three variables together → use \( (a+b+c)^2 \)

CBSE Tip

Writing the identity used earns method marks, even if final answer is wrong.

Concept Mastery Quiz

1. The degree of the constant polynomial "7" is:

A) 0

B) 1

C) Not defined

2. Which of these is a Trinomial?

A) \( x^3 \)

B) \( x^2 - 9 \)

C) \( x^2 + 5x + 6 \)

3. If \( x+y+z = 0 \), the value of \( x^3+y^3+z^3 \) is:

A) 0

B) 3xyz

C) xyz

4. The zero of the polynomial \( p(x) = 2x + 1 \) is:

A) 2

B) -1/2

C) 1/2

5. The degree of the Zero Polynomial is:

A) 0

B) 1

C) Not defined

🧠 Final Self-Check (Before Exam)

Can I identify degree instantly?
Can I reject non-polynomials quickly?
Do I know which identity to apply?
Can I find zeroes using graph idea?

If yes → you are exam-ready 🎯