Polynomials – Definition, Types, and Formulas

A polynomial is an algebraic expression built from variables, constants, and non-negative integer exponents, combined using addition, subtraction, and multiplication. The general form is P(x)=anxn+an−1xn−1+⋯+a1x+a0, where the highest exponent n is the polynomial's degree. Polynomials are classified two ways — by the number of terms (monomial, binomial, trinomial) and by degree (linear, quadratic, cubic, and higher).

The key formulas you'll meet most often are the algebraic identities — (a+b)², (a−b)², (a+b)(a−b), the cubic identities, and the Factor and Remainder theorems.

Quick Reference

What Is A Polynomial?

A polynomial is the simplest kind of algebraic expression that still does interesting work. It is made of terms, and each term is a constant multiplied by a variable raised to a whole-number power. Add or subtract terms together and you have a polynomial.

The expression 5x³−2x²+7x−4 is a polynomial in the variable x. Each term has three parts: a coefficient (the number out front), a variable, and an exponent that must be a non-negative integer. The degree of the whole polynomial is the highest exponent that appears — here, 3.

The expression x+1 is not a polynomial. The square root is the same as x¹/², and that exponent is not a whole number. Likewise 1/x is not a polynomial — it hides x⁻¹, a negative exponent.

The Types of Polynomials

There are two ways to sort polynomials, and you'll need both.

By Number of Terms

A polynomial with one term is a monomial: 7x², −3, 12xy. Two terms make a binomial: x²+5, 3x−7. Three terms make a trinomial: x²+5x+6. Four or more terms — the word polynomial covers everything beyond three terms.

By Degree

The degree is the highest exponent in the polynomial. The names map directly to it.

A constant polynomial like 7 has degree 0 because 7=7x⁰. The zero polynomial — just 0 — is a special case; it has no defined degree.

The Key Formulas

Most working with polynomials reduces to a handful of identities. Memorise the shapes; the rest follows.

Algebraic identities (binomial expansions):

(a+b)²=a²+2ab+b²

(a−b)²=a²−2ab+b²

(a+b)(a−b)=a²−b²

(a+b)³=a³+3a²b+3ab²+b³

(a−b)³=a³−3a²b+3ab²−b³

a³+b³=(a+b)(a²−ab+b²)

a³−b³=(a−b)(a²+ab+b²)

Factor theorem. If P(c)=0, then (x−c) is a factor of P(x).

Remainder theorem. When you divide P(x) by (x−c), the remainder is P(c).

The quadratic formula. For ax²+bx+c=0,

x=−b±√(b²−4ac)/2a

Working With Polynomials - Three Worked Examples

Example 1: Subtracting two polynomials

Subtract (2x²−3x+4) from (5x²+x−1).

Final answer: 3x² + 4x − 5.

Example 2: Factoring a quadratic

Factor x²+5x+6: find two numbers whose product is 6 and whose sum is 5. Those numbers are 2 and 3.

Final answer: (x+2)(x+3).

Example 3: Finding the degree of a complicated polynomial

What is the degree of 4x²y³−7xy+24? The highest among its terms is 5.

Common Polynomial Mistakes

Mistake 1: Subtracting without parentheses.

Whenever the problem says "subtract polynomial A from polynomial B" and the student writes the two expressions in a row.

Mistake 2: Adding the exponents to the coefficients.

Multiplying two monomials like 3x²⋅4x³.

Mistake 3: Confusing the degree with the number of terms.

Counting the terms instead of finding the highest exponent.

Mistake 4: Failing to combine like terms.

After expanding a product or simplifying a sum, the answer is incomplete if like terms are not combined.

Frequently Asked Questions

Is 5 a polynomial? Yes, a constant by itself is a degree-0 polynomial.

What is the degree of the zero polynomial? Undefined; some textbooks set it to −∞.

Why are negative or fractional exponents not allowed in polynomials? Because they break the closed family of polynomials.

Is there a formula for polynomials of degree five and higher? No general formula in radicals exists.

Where do polynomials show up outside maths class? Bezier curves, error-correcting codes, RSA cryptography, and more.