CBSE Class 9 Maths - Polynomials (Chapter 2)

Polynomials: Your Math Superpower! ✨

CBSE Class 9 Mathematics - Chapter 2

What is a Polynomial?

An algebraic expression made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Example: 2x² + 3x + 5

Key Components of a Polynomial

Constant: A fixed numerical value (e.g., 5, -2, ½). A constant is also considered a polynomial of degree zero.
Variable: A symbol (usually a letter like x, y, z) that represents an unknown value.
Term: Each part of the polynomial separated by a plus or minus sign.
Example: In 2x² + 3x + 5, the terms are 2x², 3x, and 5.
Coefficient: The numerical value multiplying a variable in a term.
Example: In 2x², 2 is the coefficient. In 3x, 3 is the coefficient.
Exponent (Power): The small number written above the variable indicating how many times the base is multiplied by itself. For polynomials, these must always be non-negative integers (0, 1, 2, 3...).

What is NOT a Polynomial?

An expression is NOT a polynomial if any variable has:

A negative exponent (e.g., 2x^-1)
A fractional exponent (e.g., 3&sqrt;x or 3x^1/2)
Appears in the denominator (e.g., 5/x)

Types of Polynomials (by Number of Terms)

Monomial: A polynomial with one term.
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Examples: 3x, 7, -5y²
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Binomial: A polynomial with two terms.
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Examples: x + 1, 2y² - 5
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Trinomial: A polynomial with three terms.
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Examples: x² + x + 1, a³ - 2a + 7
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Zero Polynomial: A polynomial where all coefficients are zero. Its value is always 0.
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Example: 0x² + 0x + 0 = 0
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Types of Polynomials (by Degree)

The degree of a polynomial is the highest exponent of the variable(s) in any term.

Degree of a non-zero constant (e.g., 5) is 0.
The degree of the zero polynomial (0) is undefined.

Example: Degree of 3x⁴ + 2x² + 7 is 4.
Example: Degree of 5xy³ + 2x²y² - 7x (sum of exponents in terms: 1+3=4, 2+2=4, 1) is 4.

constant polynomial: Degree 0.
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Examples: 3,  7
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Linear Polynomial: Degree 1.
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Examples: 2x + 3, y - 7
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Quadratic Polynomial: Degree 2.
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Examples: x² + 2x + 1, 5y² - 4
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Cubic Polynomial: Degree 3.
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Examples: x³ - x² + 2, 7a³ + 8
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Polynomials by Number of Variables

One Variable: Contains only one type of variable.
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Example: P(x) = x² + 2x + 1
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Two Variables: Contains two different types of variables.
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Example: P(x, y) = x² + xy + y²
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Multi-variable: Contains more than one type of variable (can be two, three, or more).

Zeros of a Polynomial

The zero(s) of a polynomial P(x) are the value(s) of the variable (say x = a) for which the value of the polynomial becomes zero, i.e., P(a) = 0.
Graphically, these are the points where the polynomial's graph crosses the x-axis.
A polynomial can have at most as many zeros as its degree.

Example: For P(x) = x - 2, the zero is found by setting P(x) = 0:

x - 2 = 0
x = 2

So, 2 is the zero of P(x) = x - 2.

Remainder Theorem

If P(x) is any polynomial of degree greater than or equal to 1, and P(x) is divided by a linear polynomial (x - a), then the remainder is P(a).
This theorem allows you to find the remainder without performing long division!

Example: Find the remainder when P(x) = x³ + 2x² - 5x + 4 is divided by (x - 1).
Here, a = 1. By Remainder Theorem, remainder = P(1).

P(1) = (1)³ + 2(1)² - 5(1) + 4
P(1) = 1 + 2 - 5 + 4
P(1) = 2

The remainder is 2.

Factor Theorem

If P(x) is a polynomial of degree n ≥ 1, then:

(x - a) is a factor of P(x) if and only if P(a) = 0. (This means a is a zero of the polynomial).
Conversely, if P(a) = 0, then (x - a) is a factor of P(x).

This theorem is crucial for factorizing polynomials!

Example: Is (x + 2) a factor of P(x) = x² + 5x + 6?
Here, x - a = x - (-2), so a = -2.
By Factor Theorem, (x + 2) is a factor if P(-2) = 0.

P(-2) = (-2)² + 5(-2) + 6
P(-2) = 4 - 10 + 6
P(-2) = 0

Since P(-2) = 0, (x + 2) is indeed a factor of x² + 5x + 6.

Laws of Exponents (Quick Review)

These rules are fundamental when working with polynomial terms!

a^m × aⁿ = a^m+n
a^m ÷ aⁿ = a^m-n (where a ≠ 0)
(a^m)ⁿ = a^mn
a⁰ = 1 (where a ≠ 0)
(ab)^m = a^mb^m
(a/b)^m = a^m/b^m (where b ≠ 0)

Important Algebraic Identities

These are equations that are true for all values of the variables. They are incredibly useful for expanding expressions, simplifying, and factorizing polynomials quickly!

Fundamental Identities:

Identity 1: (a + b)² = a² + 2ab + b²
Identity 2: (a - b)² = a² - 2ab + b²
Identity 3: a² - b² = (a - b)(a + b)
Identity 4: (x + a)(x + b) = x² + (a + b)x + ab

More Advanced Identities:

Identity 5: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Identity 6: (a + b)³ = a³ + b³ + 3ab(a + b) = a³ + 3a²b + 3ab² + b³
Identity 7: (a - b)³ = a³ - b³ - 3ab(a - b) = a³ - 3a²b + 3ab² - b³
Identity 8: a³ + b³ = (a + b)(a² - ab + b²)$
Identity 9: a³ - b³ = (a - b)(a² + ab + b²)$
Identity 10: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)$
Special Case (from Identity 10): If a + b + c = 0, then a³ + b³ + c³ = 3abc

Example: Expand (2x + 3y)² using Identity 1.
Here, a = 2x and b = 3y.

(2x + 3y)² = (2x)² + 2(2x)(3y) + (3y)²
= 4x² + 12xy + 9y²

Example: Factorize 25x² - 49 using Identity 3.
We can write this as (5x)² - (7)². Here, a = 5x and b = 7.

25x² - 49 = (5x - 7)(5x + 7)

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Tuesday, June 10, 2025