Polynomials Explained: Types, Degree and Zeroes
Polynomials are one of the most important building blocks in school algebra, and you will meet them again and again from Class 9 right up to JEE. In this guide we explain what a polynomial actually is, how to find its degree, the common types you must know, and how to calculate zeroes — all with simple worked examples.
What is a polynomial?
A polynomial is an algebraic expression made up of variables and constants, combined using only addition, subtraction and multiplication, where every power of the variable is a whole number (0, 1, 2, 3, …). For example, 3x² + 5x − 7 is a polynomial. However, expressions like 2/x, √x or x⁻¹ are not polynomials, because the exponents are not whole numbers.
Each part separated by + or − is called a term. In 3x² + 5x − 7 the terms are 3x², 5x and −7. The number multiplying a variable is the coefficient (here 3 and 5), and −7 is the constant term. If you are still getting comfortable with terms and coefficients, our algebra basics for beginners guide is a good warm-up.
Degree of a polynomial
The degree is the highest power of the variable in the polynomial. It tells you a lot about the shape and behaviour of the expression.
- 4x + 1: highest power is 1, so degree = 1.
- 2x² − 3x + 6: highest power is 2, so degree = 2.
- x³ − 5: highest power is 3, so degree = 3.
- 7 (a constant): degree = 0, because 7 = 7x⁰.
Types of polynomials by degree
Polynomials are named by their degree, and these names appear constantly in CBSE and ICSE textbooks.
- Linear polynomial (degree 1): e.g. 2x + 3. Its graph is a straight line.
- Quadratic polynomial (degree 2): e.g. x² − 5x + 6. Its graph is a parabola.
- Cubic polynomial (degree 3): e.g. x³ − 6x² + 11x − 6.
We can also classify by the number of terms: a monomial has one term (5x²), a binomial has two (x + 4), and a trinomial has three (x² + 2x + 1).
Zeroes of a polynomial
A zero (or root) of a polynomial p(x) is a value of x that makes p(x) = 0. Geometrically, a zero is where the graph cuts the x-axis.
Linear example: Find the zero of p(x) = 2x + 6. Set 2x + 6 = 0, so 2x = −6, giving x = −3. So the only zero is −3.
Quadratic example: Find the zeroes of p(x) = x² − 5x + 6. Factorise: x² − 5x + 6 = (x − 2)(x − 3). Setting each factor to zero gives x = 2 and x = 3. You can verify: 2² − 5(2) + 6 = 4 − 10 + 6 = 0. A key fact is that a quadratic has at most two zeroes, and in general a polynomial of degree n has at most n zeroes. To master these techniques, see our step-by-step guide on how to solve quadratic equations.
Relationship between zeroes and coefficients
For a quadratic ax² + bx + c, if the zeroes are α and β, then α + β = −b/a and αβ = c/a. For x² − 5x + 6 we have α + β = 5 and αβ = 6, matching zeroes 2 and 3 exactly. This idea is heavily tested in board exams and is developed further in Class 10 Maths, while the foundations start in Class 9 Maths.
Frequently asked questions
What is the degree of a zero polynomial?
The zero polynomial (the constant 0) has no defined degree, because it has no highest power term. This is different from a non-zero constant like 7, whose degree is 0.
Can a polynomial have negative or fractional powers?
No. By definition, every exponent in a polynomial must be a non-negative whole number. Expressions like x⁻² or x^(1/2) are algebraic expressions but not polynomials.
How many zeroes does a cubic polynomial have?
A cubic polynomial (degree 3) has at most three real zeroes. It always has at least one real zero, and the total number of zeroes counted with complex roots is exactly three.