What is Polynomial Class 10 in Hindi: Complete NCERT Guide

Definition of Polynomial in Class 10 NCERT Mathematics

In Class 10 NCERT Mathematics, a polynomial is defined as an algebraic expression consisting of variables and coefficients, involving only addition, subtraction, and multiplication. Variables have whole number exponents (non-negative integers).

General form:

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

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

For example, $3x^2 + 5x - 7$ is a polynomial of degree 2.

Types of Polynomials Based on Degree and Terms

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

Understanding these types helps in solving polynomial-related problems efficiently.

Important Properties and Operations on Polynomials

Polynomials follow specific algebraic rules:

• Addition: Combine like terms by adding coefficients.
• Subtraction: Subtract coefficients of like terms.
• Multiplication: Multiply each term of one polynomial by every term of the other.

Example:

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

$$ (3x^2 + 2x + 1) + (x^2 - x + 4) = (3x^2 + x^2) + (2x - x) + (1 + 4) = 4x^2 + x + 5 $$

• Degree of Polynomial: Highest power of variable with non-zero coefficient.
• Zero Polynomial: All coefficients zero; degree is undefined or sometimes taken as -1.

These properties are essential for solving NCERT exercises and CBSE exam questions.

How to Identify a Polynomial: Key Points for Class 10 Students

To identify if an expression is a polynomial, check the following:

• Variables have whole number exponents only (no negative or fractional powers).
• No variables in the denominator.
• No variables inside roots or absolute value.
• Only addition, subtraction, and multiplication operations are allowed.

Examples:

• $4x^3 + 3x - 5$ is a polynomial.
• $\frac{1}{x} + 2$ is not a polynomial (variable in denominator).
• $\sqrt{x} + 1$ is not a polynomial (variable under root).

This understanding helps in quickly solving polynomial questions in Class 10 exams.

Worked Example: Finding Degree and Type of Polynomial

Example: Determine the degree and type of the polynomial $5x^4 - 3x^2 + 7x - 9$.

Solution:

• Highest power of $x$ is 4, so degree = 4 (Quartic polynomial).
• Number of terms = 4, so it is a polynomial with four terms (not monomial/binomial/trinomial).

Summary:

• Degree: 4
• Type by degree: Quartic
• Type by terms: Polynomial with four terms

This method is useful for classification questions in NCERT Class 10 exercises.

Comparison Table: Polynomial vs Non-Polynomial Expressions

Here's a quick comparison to help distinguish polynomials from non-polynomials:

| Variables' exponents | Whole numbers (0,1,2,...) | Negative, fractional, or variable in denominator

| Operations allowed | Addition, subtraction, multiplication | Division by variable, roots, absolute values

| Example | $3x^2 + 2x + 1$ | $\frac{1}{x} + 4$, $\sqrt{x} + 3$

This table clarifies common doubts for Class 10 students preparing for CBSE exams.