Introduction to Polynomials

What is a Polynomial?

The chapter Polynomials and Factorisation begins by introducing one of the most important building blocks of algebra in Class 9 Mathematics across CBSE, Telangana, and Andhra Pradesh syllabi. Students already work with algebraic expressions such as πr², lbh, and x² − 2x + 1, and this lesson formally defines what makes an expression a polynomial.

An algebraic expression is called a polynomial only when every variable in it has a non-negative integer as its exponent. This means powers like x², x³, or x⁰ are allowed, but expressions involving negative exponents or roots of variables are not polynomials.

Interestingly, even a constant like 5 qualifies as a polynomial, because it can be written as 5x⁰, and 0 is a non-negative integer.

Identifying Polynomials and Non-Polynomials

A key skill developed in this section is checking whether a given expression is a polynomial by examining the exponents carefully. This is tested through several examples that students should practice identifying.

• Polynomials: 4x² + 5x − 2, y² − 8, the constant 5, 3x² + 5y, and 3xyz — all variables have whole-number exponents.
• Not polynomials: 2x² + 3/x − 5 and 1/x + 1 — these contain terms with negative exponents (x⁻¹).
• Not polynomials: √x — here the exponent is x^(1/2), which is not an integer.

Understanding why each expression qualifies or fails to qualify builds a strong foundation for recognising polynomials quickly during exams.

Polynomials in One Variable

A variable is a symbol — usually x, y, or z — that can represent any real number. When an algebraic expression contains only one such symbol, it is called a polynomial in one variable. Expressions like 3x² − 2x + 1, x⁴ − x² + 5x − 5, and x³ − x² are common examples studied in this lesson.

A useful real-life illustration is the formula for the perimeter of a square, written as 4l, where l represents the side length. Here, 4 is a fixed constant while l is the variable whose value changes depending on the square being measured. When the constant itself is unknown, letters such as a, b, or c are used to represent it, giving general expressions like ax, by, or cz.

Degree of a Polynomial

The degree of a term is found by adding up the exponents of all its variables, while the degree of the entire polynomial is the highest degree among its terms. This concept is essential for classifying polynomials later in the chapter.

A general polynomial in one variable x of degree n is written in the standard form below, where a₀ cannot be zero.

Special cases worth remembering include that the constant term −7 has degree 0 (since it equals −7x⁰), while the zero polynomial (the number 0) has an undefined degree, since 0 can be written as 0×x raised to any power.

Classifying Polynomials by Degree and Number of Terms

Polynomials are named based on two different criteria — their degree and the number of non-zero terms they contain. Knowing these names helps students communicate clearly about polynomials in higher classes, especially when studying quadratic equations and cubic polynomials in Class 10.

• By degree: constant (degree 0), linear (degree 1), quadratic (degree 2), and cubic (degree 3) polynomials.
• By number of terms: monomial (1 term), binomial (2 terms), trinomial (3 terms), and multinomial (more than 3 terms).

Polynomials in one variable are commonly represented using function notation, such as p(x), q(z), or f(y), making it easier to refer to them in equations and problem statements.

What This Lesson Prepares You For

This introductory lesson sets the stage for the rest of the chapter, where students learn to find zeroes of a polynomial, apply the Remainder Theorem, and explore factorisation techniques in depth. A strong grip on identifying polynomials and their degrees here makes it much easier to follow Exercise 2.1 on zeroes of polynomials and later topics on factorisation methods. These ideas also connect directly to Class 10 topics such as the relationship between zeroes and coefficients, which build on the definitions introduced here.