What Is Polynomial Class 9 With Example: Complete Guide

Definition of Polynomial in Class 9 Mathematics

A polynomial is an algebraic expression consisting of variables and coefficients, combined using only addition, subtraction, and multiplication. The variables have non-negative integer exponents.

Formally, a polynomial in one variable $x$ can be written as:

$$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$

where:

• $a_n, a_{n-1}, \dots, a_0$ are constants called coefficients
• $n$ is a non-negative integer called the degree of the polynomial

Example:

• $5x^3 - 2x^2 + 3x - 7$ is a polynomial of degree 3
• $4x + 9$ is a polynomial of degree 1

Polynomials do not include variables with negative or fractional powers or variables in denominators.

Types of Polynomials With Examples

Polynomials are classified based on the number of terms they contain:

Each term in a polynomial is called a monomial. The degree of a polynomial is the highest power of the variable in any term.

Example:

• $6x^4 + 3x^3 - x + 8$ is a polynomial with 4 terms and degree 4.

Understanding these types helps in identifying and working with polynomials in Class 9 NCERT exercises.

Degree and Coefficients of a Polynomial Explained

The degree of a polynomial is the highest exponent of the variable in the expression.

• For $P(x) = 4x^5 + 2x^3 - x + 7$, degree = 5
• For $Q(x) = 3x^2 + 6x + 9$, degree = 2

The coefficient is the numerical factor multiplied by the variable term.

• In $5x^3$, coefficient = 5
• In $-2x^2$, coefficient = -2

The constant term (without variable) is also considered a coefficient of $x^0$.

Worked Example: Find the degree and coefficients of $7x^4 - 3x^2 + x - 9$.

• Degree: 4 (highest power is 4)
• Coefficients: 7, -3, 1, -9

Knowing degree and coefficients is crucial for solving polynomial problems in Class 9.

How to Identify Polynomial Expressions: Key Rules

To check if an expression is a polynomial, follow these rules:

• Variables must have whole number exponents (0, 1, 2, ...).
• No variables in denominators.
• No negative or fractional powers of variables.
• Only addition, subtraction, and multiplication are allowed.

Examples:

• $x^3 + 4x - 7$ is a polynomial.
• $\frac{1}{x} + 5$ is not a polynomial (variable in denominator).
• $x^{-2} + 3$ is not a polynomial (negative exponent).
• $\sqrt{x} + 2$ is not a polynomial (fractional power).

This understanding helps avoid common mistakes in Class 9 NCERT exercises.

Addition and Subtraction of Polynomials: Simple Steps

Adding or subtracting polynomials involves combining like terms — terms with the same variable and exponent.

Steps: 1. Arrange terms in descending order of degree. 2. Combine coefficients of like terms. 3. Write the simplified polynomial.

Example 1: Add $P(x) = 3x^2 + 5x - 2$ and $Q(x) = x^2 - 3x + 4$.

$$ P(x) + Q(x) = (3x^2 + 5x - 2) + (x^2 - 3x + 4) = (3x^2 + x^2) + (5x - 3x) + (-2 + 4) = 4x^2 + 2x + 2 $$

Example 2: Subtract $Q(x)$ from $P(x)$:

$$ P(x) - Q(x) = (3x^2 + 5x - 2) - (x^2 - 3x + 4) = (3x^2 - x^2) + (5x + 3x) + (-2 - 4) = 2x^2 + 8x - 6 $$

Practice these operations to strengthen your polynomial skills for Class 9 exams.

Multiplication of Polynomials: Basics and Examples

Multiplying polynomials involves multiplying each term of one polynomial by every term of the other, then combining like terms.

Example: Multiply $(x + 3)$ and $(x^2 + 2x + 1)$.

Step 1: Multiply each term in the first polynomial by each term in the second:

• $x \times x^2 = x^3$
• $x \times 2x = 2x^2$
• $x \times 1 = x$
• $3 \times x^2 = 3x^2$
• $3 \times 2x = 6x$
• $3 \times 1 = 3$

Step 2: Combine like terms:

$$ x^3 + 2x^2 + x + 3x^2 + 6x + 3 = x^3 + (2x^2 + 3x^2) + (x + 6x) + 3 = x^3 + 5x^2 + 7x + 3 $$

This method applies to all polynomial multiplication problems in Class 9 NCERT.

Common Mistakes to Avoid While Working With Polynomials

When studying polynomials in Class 9, avoid these common errors:

• Treating variables with negative or fractional powers as polynomials.
• Forgetting to combine like terms after addition or subtraction.
• Mixing up degree and coefficient definitions.
• Ignoring zero coefficients (terms with zero coefficient do not appear).
• Incorrectly multiplying terms without applying distributive property.

Careful practice and understanding of concepts will help you avoid these mistakes in exams.