Polynomial and Examples- Complete Guide

What Is a Polynomial?

A polynomial is a mathematical expression with variables and coefficients. You combine terms using addition, subtraction, and multiplication. No division by variables. No negative or fractional exponents. That's it.

Here's what a polynomial looks like:

3x² + 2x - 5

And here's one that's not a polynomial:

3x⁻² + 2/x

Negative exponents and variables in denominators disqualify an expression. Keep this in mind when your textbook tries to trick you.

The Parts of a Polynomial

Every polynomial has terms. Each term has a coefficient and a variable part.

In 4x³ - 7x² + x - 3:

• 4x³ — coefficient is 4, variable part is x³
• -7x² — coefficient is -7, variable part is x²
• x — coefficient is 1 (it's just x, which means 1x)
• -3 — this is a constant term, no variable

The constant term is technically a coefficient too. It's the coefficient of x⁰.

Types of Polynomials by Number of Terms

Classifying by terms is straightforward:

• Monomial — one term (5x², -3y, 7)
• Binomial — two terms (x + 2, 4x² - 9)
• Trinomial — three terms (x² + 3x + 2)

Beyond three terms, people just say "polynomial." Nobody calls something a "quadnomial."

Types of Polynomials by Degree

The degree is the highest exponent on any variable. This matters more than you think.

Common Degree Names

• Degree 0 — constant (5, -12)
• Degree 1 — linear (x + 3, 2y - 1)
• Degree 2 — quadratic (x² + x + 1)
• Degree 3 — cubic (x³ - 4x² + 2x + 7)
• Degree 4 — quartic
• Degree 5 — quintic

After quintic, mathematicians stop naming them. You're on your own for degree 6 and beyond.

Finding the Degree With Multiple Variables

When you have multiple variables, find the degree of each term by adding the exponents. The degree of the polynomial is the highest sum.

4x³y² + 3x²y⁴ - 7xy

• First term: 3 + 2 = 5
• Second term: 2 + 4 = 6
• Third term: 1 + 1 = 2

This polynomial has degree 6.

Evaluating Polynomials

To evaluate means to substitute a number for the variable and calculate the result.

Example: Evaluate 2x² - 3x + 1 when x = 4

Step 1: Replace x with 4

2(4)² - 3(4) + 1

Step 2: Follow order of operations — exponents first

2(16) - 3(4) + 1

Step 3: Multiply

32 - 12 + 1

Step 4: Add and subtract

21

That's it. Plug in, simplify. Nothing fancy.

Adding and Subtracting Polynomials

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

Example: Add (3x² + 2x - 5) + (x² - 4x + 3)

Drop the parentheses and group like terms:

3x² + x² + 2x - 4x - 5 + 3

Combine:

4x² - 2x - 2

Subtraction works the same way, but distribute the negative sign first.

Example: Subtract (2x² + 5x - 1) - (x² - 3x + 4)

Distribute the negative:

2x² + 5x - 1 - x² + 3x - 4

Combine like terms:

x² + 8x - 5

Multiplying Polynomials

Use the distributive property. Multiply every term in the first polynomial by every term in the second polynomial.

Multiplying a Monomial by a Polynomial

Example: 3x(2x² - 4x + 5)

Multiply 3x by each term:

3x · 2x² = 6x³

3x · (-4x) = -12x²

3x · 5 = 15x

Result: 6x³ - 12x² + 15x

Multiplying Two Binomials (FOIL)

FOIL stands for First, Outer, Inner, Last. It's a mnemonic, not a new rule.

Example: (x + 3)(x - 5)

• First: x · x = x²
• Outer: x · (-5) = -5x
• Inner: 3 · x = 3x
• Last: 3 · (-5) = -15

Combine: x² - 5x + 3x - 15 = x² - 2x - 15

Multiplying Larger Polynomials

When neither polynomial is a binomial, just multiply everything by everything.

Example: (x + 2)(x² - x + 3)

Multiply x by each term, then 2 by each term:

x³ - x² + 3x + 2x² - 2x + 6

Combine like terms:

x³ + x² + x + 6

Dividing Polynomials

Polynomial long division is tedious. Synthetic division is faster but only works when dividing by a linear expression in the form (x - c).

Polynomial Long Division

Divide x² + 5x + 6 by (x + 2).

Step 1: How many times does x go into x²? x times. Write x above the division bar.

Step 2: Multiply x by (x + 2) = x² + 2x. Subtract from the original.

Step 3: Bring down the next term. You have 3x + 6.

Step 4: How many times does x go into 3x? 3 times. Write 3 above.

Step 5: Multiply 3 by (x + 2) = 3x + 6. Subtract. Remainder is 0.

Result: x + 3

Synthetic Division

Divide x² + 5x + 6 by (x - 2). The divisor is x - 2, so c = 2.

Write the coefficients: 1, 5, 6

Bring down the 1. Multiply by 2, write under 5. Add. Multiply result by 2. Add.

Result: x + 7 remainder 20

Check: (x + 7) + 20/(x - 2)

Factoring Polynomials

Factoring means rewriting as a product of simpler polynomials. This is essential for solving equations.

Factoring Out the Greatest Common Factor (GCF)

Find what divides every term.

Example: Factor 6x³ + 9x² - 3x

GCF is 3x:

3x(2x² + 3x - 1)

That's the factored form.

Factoring Trinomials

For x² + bx + c, find two numbers that multiply to c and add to b.

Example: Factor x² + 5x + 6

Find two numbers that multiply to 6 and add to 5: 2 and 3.

Result: (x + 2)(x + 3)

For 2x² + 7x + 3, you need two numbers that multiply to (2)(3) = 6 and add to 7: 6 and 1.

Rewrite 7x as 6x + x, then factor by grouping.

Difference of Squares

a² - b² = (a + b)(a - b)

Example: x² - 16

x² - 16 = x² - 4² = (x + 4)(x - 4)

Sum and Difference of Cubes

• a³ + b³ = (a + b)(a² - ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)

These formulas are worth memorizing. You'll see them repeatedly.

Polynomial Zeros (Roots)

The zeros of a polynomial are the x-values that make the polynomial equal to zero. They're also called roots.

If (x - 3) is a factor, then x = 3 is a zero.

Example: Find the zeros of x² - x - 6

Factor: (x - 3)(x + 2)

Set each factor to zero:

• x - 3 = 0 → x = 3
• x + 2 = 0 → x = -2

Zeros are 3 and -2.

The Fundamental Theorem of Algebra

A polynomial of degree n has exactly n complex roots (counting multiplicities and complex numbers).

A quadratic (degree 2) has 2 roots. A cubic (degree 3) has 3 roots. This is non-negotiable.

Comparing Polynomial Operations

Getting Started: How to Practice

If you want to actually get good at polynomials, here's what to do:

• Start with evaluating expressions — plug in values until it becomes automatic
• Move to combining like terms — this is the foundation for everything else
• Practice multiplication with increasingly larger polynomials
• Master factoring trinomials before touching harder methods
• Use the quadratic formula when factoring gets messy

Don't skip steps. Students who try to do polynomial division without solid factoring skills struggle. Build the foundation first.

Quick Reference: Common Mistakes

• Forgetting to distribute the negative when subtracting
• Combining unlike terms (x² + x doesn't equal x³)
• Losing track of negative signs when factoring
• Forgetting that 0 is a valid root
• Assuming a polynomial has only real roots — complex numbers exist

Polynomials are algebraic building blocks. Once you understand operations, factoring, and zeros, you're set up for success with quadratic equations, rational functions, and calculus. Master these basics or regret it later.