Exponents and Polynomials- Definitions

What Are Exponents and Polynomials?

Exponents and polynomials are the building blocks of algebra. You cannot solve equations, factor expressions, or graph curves without understanding these two concepts first.

This guide cuts through the fluff and gives you the actual definitions you need to work with these mathematical objects. Nothing else.

Exponents: The Basics

An exponent tells you how many times to multiply a number by itself. It's written as a small number to the upper right of a base.

Exponent Definition

For any base number a and positive integer n:

aⁿ = a × a × a × ... × a (n times)

Example: 2³ = 2 × 2 × 2 = 8

The number 2 is the base. The number 3 is the exponent. Simple.

Special Exponent Rules

• Zero exponent: a⁰ = 1 (for any non-zero a)
• Negative exponent: a⁻ⁿ = 1/aⁿ
• Product rule: a^m × aⁿ = a^m+n
• Quotient rule: a^m ÷ aⁿ = a^m−n
• Power of a power: (a^m)ⁿ = a^mn

Fractional Exponents

Fractional exponents represent roots. The denominator becomes the root.

a^1/2 = √a

a^1/3 = ∛a

a^2/3 = (∛a)²

Polynomials: The Definition

A polynomial is a sum of terms, where each term is a constant multiplied by a power of a variable. The exponents must be whole numbers (0, 1, 2, 3...).

General Form

P(x) = a_nxⁿ + a_n−1xⁿ⁻¹ + ... + a₂x² + a₁x + a₀

The letters a_n, a_n−1, etc., are coefficients. They can be any real number.

What Makes Something a Polynomial?

• Variables have non-negative integer exponents only
• No variables in denominators
• No variables under radicals
• Finite number of terms

Examples of polynomials: 3x² + 2x − 5, 4y³ − y + 7, 12

Not polynomials: 1/x, √x, x⁻², sin(x)

Types of Polynomials

Polynomials are classified by the number of terms they contain.

• Monomial: one term (e.g., 5x³)
• Binomial: two terms (e.g., x² + 3x)
• Trinomial: three terms (e.g., x² − 4x + 4)
• Multinomial: four or more terms

Classification by Degree

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

Degree of a Polynomial with Multiple Variables

When a polynomial has more than one variable, the degree is the sum of the exponents in the highest term.

Example: 3x³y² + 2xy⁴

First term degree: 3 + 2 = 5

Second term degree: 1 + 4 = 5

Overall degree: 5

Terms, Coefficients, and Constants

Take the polynomial: 4x³ + 3x² − 7x + 2

• Terms: 4x³, 3x², −7x, 2
• Coefficients: 4, 3, −7 (the numbers multiplying the variables)
• Constant term: 2 (the term with no variable)
• Leading coefficient: 4 (coefficient of the highest-degree term)

Evaluating Polynomials

To evaluate a polynomial, substitute a number for the variable and simplify.

Example: P(x) = x² + 3x − 4

Find P(2):

P(2) = (2)² + 3(2) − 4

P(2) = 4 + 6 − 4

P(2) = 6

Adding and Subtracting Polynomials

Combine like terms. Like terms have the same variable raised to the same power.

Example: (3x² + 2x + 1) + (x² − 3x + 4)

= 3x² + x² + 2x − 3x + 1 + 4

= 4x² − x + 5

Multiplying Polynomials

Multiplying a Monomial by a Polynomial

Distribute the monomial to every term.

Example: 3x(2x² − 4x + 5)

= 3x · 2x² − 3x · 4x + 3x · 5

= 6x³ − 12x² + 15x

Multiplying Two Binomials (FOIL)

Use FOIL: First, Outer, Inner, Last

Example: (x + 3)(x + 2)

= x·x + x·2 + 3·x + 3·2

= x² + 2x + 3x + 6

= x² + 5x + 6

Multiplying Any Two Polynomials

Multiply every term in the first polynomial by every term in the second polynomial. Then combine like terms.

Common Factor vs. Greatest Common Factor

Common factor: any factor shared by all terms

Greatest common factor (GCF): the largest factor shared by all terms

Example: 12x³ + 18x²

GCF = 6x²

Factored form: 6x²(2x + 3)

Quick Reference Table: Exponent Rules

Getting Started: How to Identify Polynomials

Follow these steps to check if an expression is a polynomial:

1. Look at all exponents on variables
2. Verify every exponent is a non-negative integer (0, 1, 2, 3...)
3. Check that no variable appears in a denominator
4. Confirm no variable appears under a radical
5. If all checks pass, you have a polynomial

Practice identifying these:

• 5x⁴ + 2x − 3 ✓ Polynomial
• 1/x² ✗ Not a polynomial
• √x + 5 ✗ Not a polynomial
• 7x³ − 2x⁰ ✓ Polynomial (x⁰ = 1, so this is just a constant)

What's Next?

Once you understand these definitions, you need to learn:

• Factoring polynomials (finding what multiplies to give you the original)
• Solving polynomial equations by setting them equal to zero
• Dividing polynomials using long division or synthetic division

These definitions are your foundation. Everything else in algebra builds on them.