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. It does not include division by variables or negative powers.

Formally, a polynomial in one variable $x$ looks like:

$$a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$

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: $3x^2 + 5x - 7$ is a polynomial of degree 2.

This definition aligns with the Class 9 NCERT syllabus and forms the foundation for understanding polynomials.

Types of Polynomials with Examples

Polynomials are classified based on the number of terms and the degree:

Degree-based classification:

• Degree 0: Constant polynomial, e.g., $5$
• Degree 1: Linear polynomial, e.g., $4x + 1$
• Degree 2: Quadratic polynomial, e.g., $x^2 + 3x + 2$

Understanding these types helps in solving polynomial problems in Class 9 exams.

How to Find the Degree of a Polynomial

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

Steps to find degree:

1. Identify each term's exponent of the variable. 2. Find the term with the largest exponent. 3. The exponent of that term is the degree.

Example: For $4x^3 + 2x^5 - 7x + 9$, the degrees of terms are 3, 5, 1, and 0 respectively.

• Highest degree is 5
• So, degree of polynomial is 5

Note: For polynomials in more than one variable, degree is the highest sum of exponents in any term.

Addition and Subtraction of Polynomials: Simple Rules

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

Example:

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

$$egin{align} P(x) + Q(x) &= (3x^2 + 5x - 2) + (x^2 - 3x + 4) \\ &= (3x^2 + x^2) + (5x - 3x) + (-2 + 4) \\ &= 4x^2 + 2x + 2 \end{align}$$

Subtraction example:

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

$$egin{align} P(x) - Q(x) &= (3x^2 + 5x - 2) - (x^2 - 3x + 4) \\ &= 3x^2 - x^2 + 5x + 3x - 2 - 4 \\ &= 2x^2 + 8x - 6 \end{align}$$

Practice such problems to score well in Class 9 NCERT exams.

Multiplication of Polynomials with Worked Example

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

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

$$egin{align} (x + 3)(x^2 + 2x + 1) &= x(x^2 + 2x + 1) + 3(x^2 + 2x + 1) \\ &= (x^3 + 2x^2 + x) + (3x^2 + 6x + 3) \\ &= x^3 + (2x^2 + 3x^2) + (x + 6x) + 3 \\ &= x^3 + 5x^2 + 7x + 3 \end{align}$$

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

Common Mistakes to Avoid While Studying Polynomials

• Confusing degree with coefficient
• Adding unlike terms instead of like terms
• Forgetting to apply the distributive property in multiplication
• Ignoring zero coefficients
• Misinterpreting negative signs

Always double-check terms and powers during calculations. Practicing NCERT exercises thoroughly helps avoid these errors and improves exam performance.