Polynomials Notes for Class 9

A polynomial is an algebraic expression that consists of one or more terms. Each term is made up of a constant multiplied by a variable raised to a whole number power.
Example: 2x² + 3x + 5 is a polynomial.
Here, 2x², 3x, and 5 are the terms of the polynomial. In each term, the numbers 2, 3, and 5 are called coefficients, and x is the variable. The powers of x (2 and 1) are whole numbers, which makes it a polynomial.

If the powers of variables are fractions or negative, then the expression is not a polynomial. For example, x^-1 + 2 or √x + 3 are not polynomials because the powers are not whole numbers.

Polynomials can have one or more variables, but in Class 9, we mainly study polynomials in one variable, usually written as x.

Types of Polynomials

Polynomials are classified in two ways — based on the number of terms and based on the degree.

Based on the Number of Terms

• Monomial: A polynomial with only one term. Example: 3x, 5y², -7.
• Binomial: A polynomial with two terms. Example: x + 2, 2x² + 3x.
• Trinomial: A polynomial with three terms. Example: x² + 3x + 2.

Based on the Degree

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

• Linear Polynomial: The highest power of the variable is 1. Example: 2x + 3.
• Quadratic Polynomial: The highest power is 2. Example: x² + 2x + 1.
• Cubic Polynomial: The highest power is 3. Example: x³ – 3x + 2.

Constant Polynomials

A constant polynomial is a polynomial that has no variable. It consists of only one number.
Example: 3, -5, or 12.
Since there is no variable, the degree of a constant polynomial is zero. For example, in the polynomial 7, we can also write it as 7x⁰ because any number raised to the power 0 is 1.

Zero Polynomial

A zero polynomial is a special polynomial in which all coefficients are zero.
Example: 0x² + 0x + 0 = 0.
It is called a zero polynomial because it gives zero for all values of x. It has no defined degree.

What are Constants and Variables?

A constant is a fixed value that does not change. Examples of constants are numbers like 2, -5, or 7.
A variable is a symbol, such as x, y, or z, which can take different values.
Example: In the expression 4x² + 3x + 2,

• 4 and 3 are constants (coefficients).
• x is the variable.

Constants and variables together form the terms of a polynomial. For instance, in 6x², 6 is the constant part and x² is the variable part.