Class 9 - Polynomials in One Variable | Polynomials Maths

Polynomials in one variable refers to the algebraic expressions that contain only one variable along with coefficients and constant terms. For example, 4x2 + 8x + 6 is a polynomial in one variable that is x. A variable in an algebraic expression is a term that does not have a fixed value and can take a real value. Understanding variables, constants, terms, coefficients and exponents are important to identify and solve the algebraic problems based on the topic of polynomials in one variable. On this page we will cover in detail about polynomials in one variable including terms, degree, factors and coefficients.

Table of Contents

• What are Polynomials in One Variable
• Terms in Polynomials
• Degree of Polynomials
• Polynomial Factors
• Solved Examples on Polynomials in One Variables

What are Polynomials in One Variable

Polynomials in one variable refers to an algebraic expression that contains only one variable. I.e., x or y.

The standard form to express polynomials in one variables is: P(x) = an xn + an-1 xn - 1 + an-2 xn-2 + ... + ai x + a0

In this polynomial x is the only variable, so it is a polynomial in one variable.

For instance, x3 – x2 + 4x + 7 is a polynomial in x. In the above polynomial x is the only variable, so it is a polynomial in one variable.

Expressions of these forms are called polynomials in one variable. In the example

above, the variable is x. Similarly, 3y2 + 5y is a polynomial in the variable y and t2 + 4 is a polynomial in the variable t.

Here are some key terms related to polynomials:

• Variables: A variable in an algebraic expression is a term that does not have a fixed value and can change over time. For example, in polynomial x2 + 4, there are two terms x is a variable and 4 is a constant.
• Constants: Constant refers to the terms that have a fixed value. For example, in expression 3x + 5, numerical terms ‘5’ is a constant as it holds a constant value and is not equal to zero.
• Coefficient: The number written with a variable is called its coefficient. For example, in expression 3x + 5, 3 is the coefficient of variable x
• Leading coefficient: The coefficient of the term with the highest degree is called a leading coefficient. For example, in the polynomial 3y2 + 5y + 7, the leading coefficient is 3 as it has the term with highest variable power. I.e., 3y2

Some other terms related to polynomials in one variable are explained below in detail.

Read more: Quadratic Equations

Terms in Polynomials

In the polynomial x2+ 2x, the expressions x2 and 2x are called the terms of the polynomial. Similarly, the polynomial 3y2 + 5y + 7 has three terms, namely, 3y2, 5y and7.

Degree of Polynomials

The degree of a polynomial is the highest power of the exponent. For example, in polynomial x2 + 4x + 7 the highest power of the exponent is the degree of the polynomial, that is, 2.

Polynomial Factors

Factoring polynomials means expressing the polynomial as a product of two or more simpler polynomials. For example, the polynomial expression: x² + 5x + 6 has 2 factors: (x + 2) and (x + 3). When multiplied these factors give the same polynomial: (x + 2)(x + 3) = x² + 5x + 6

Solved Examples on Polynomials in One Variables

Example 1: How many variables and terms are there in the polynomial: x3 + 4x² + 7x – 2?

Solution: The given polynomial x3 + 4x² + 7x – 2 has only one variable x.

This polynomial has 4 terms, namely, x3 , 4x² , 7x and –2.

Example 2: Find the degree of each of the polynomials: x5– x4 + 3

Solution: The highest power of the variable is 5. So, the degree of the polynomial

is 5.

Example 3: What is the leading coefficient of the polynomial P(x)=2 – y5 – y3 + 2y⁸

Solution: The highest power of the variable is 8. So, the leading coefficient of the polynomial is 2 which is coefficient for the term with the highest power.

Example 4: What is the degree of polynomial p(x)=2?

Solution: The only term here is 2 which can be written as 2x0. So the exponent of x is 0.

Therefore, the degree of the polynomial is 0.