Polynomials Explained: Definition, Degree & Factoring

What Is a Polynomial?

A polynomial is a mathematical expression made of variables, numbers, and whole-number exponents. These terms are combined with plus and minus signs. You can add, subtract, and multiply the parts, but you can’t divide by a variable. The variable is sometimes called an indeterminate.

For example, 2x² – 7x + 6 is a polynomial with one variable. It has three terms: 2x², -7x, and 6.

The word “polynomial” comes from Greek: “poly” means many, and “nomial” means terms. So these expressions are made of many terms—but not an infinite number.

Polynomials follow specific rules that make them easy to work with. You can add or multiply them and still end up with another polynomial. Also, if they have only one variable, they produce smooth and continuous curves when you graph them.

In this post, you will learn what a polynomial is, how to identify and classify them, how to perform operations like addition, subtraction, multiplication, and division, and how to factor and solve polynomial expressions. You’ll also explore advanced topics like the degree of polynomial functions, and theorems such as the Remainder Theorem, Factor Theorem, and Descartes’ Rule of Signs.

Polynomial Definition

Each piece of a polynomial is called a term and the expression separates them by plus or minus signs.

Each term can include:

• A constant (like 3 or -7)
• A variable (like x, y, or z)
• An exponent (like the 2 in x²)

For example, in 3x² + 2x – 8, there are three terms: 3x², 2x, and -8.

To be a polynomial, an expression must meet all these conditions:

• The exponents on the variables must be whole numbers (0, 1, 2, 3, …).
  That means no negative exponents and no fractions or square roots in the exponent.
• The expression uses only addition, subtraction, and multiplication. You cannot divide by variables (no variables in the denominator).
• The coefficients (numbers in front of the variables) can be any real number, including fractions, decimals, and square roots.

If an expression follows all these rules, it’s a polynomial.

Polynomial or Not?

These are valid polynomials:

• 4x² + 3x – 5 → valid (all rules followed)
• x/2 → valid (dividing by a number is okay)
• √3x² → valid (√3 is just a constant even though it is an irrational number)
• 2x
• x – 4
• -3x²y + 5
• 6 (a constant is still a polynomial!)

These are not polynomials:

• x^-2 (negative exponent)
• 1/x (division by a variable)
• √x (fractional exponent)

Degree of a Polynomial

The degree is the term with the highest exponent in the polynomial.

Polynomial	Degree	Example
Constant	0	7
Linear	1	4x + 2
Quadratic	2	x2 – 3x + 1
Cubic	3	2x3 + x
Quartic	4	x4 + 5x2 – 6

To find the degree, look for the term with the largest exponent.

Note: Polynomials are a great way to fit curves in regression analysis. The number of bends in the fitted line is the degree minus 1. For example, a regression model with a quadratic term (degree = 2) fits data with one bend. Cubic polynomial have multiple bends, allowing the to have local minima and local maxima.

Types by Number of Terms

• Monomial: 1 term (e.g., 5x³, 8, or -2xy)
• Binomial: 2 terms (e.g., x² + 3, 6a⁴ – 9)
• Trinomial: 3 terms (e.g., 2x² + 3x – 1)
• 4+ terms: Just called polynomials

The names are based on how many terms the expression has.

Standard Form of a Polynomial

Polynomials are often written in standard form, with the terms ordered from the highest degree to the lowest. This format makes them easier to read and work with.

For example, let’s write 4x² – 6 + 3x⁵ + x in standard form.

The degree is highest in the 3x⁵ term, so list it first. Then list the remaining terms in descending order of exponents. Be sure to retain the correct signs for each term as you rearrange them in the standard form.

3x⁵ + 4x² + x – 6

Mathematical Operations

You can add, subtract, multiply, and sometimes divide polynomials.

Polynomials follow the usual rules of arithmetic, so you’ll use operations like addition, subtraction, multiplication, and sometimes division. If you need a refresher on the correct order of operations, read my post on PEMDAS the mathematical order of operations.

Here’s how each operation works.

Addition of Polynomials

To add polynomials, group and combine like terms—those with the same variable and exponent.

Example:
Add 4x² + 3x – 5 and 2x² – x + 7

Solution:
Group like terms:
(4x² + 2x²) + (3x – x) + (-5 + 7)

Combine:
6x² + 2x + 2

Polynomial Subtraction

To subtract, change the sign of each term in the second polynomial, then combine like terms.

Example:
Subtract 2x² – x + 7 from 4x² + 3x – 5

Solution:
(4x² + 3x – 5) – (2x² – x + 7)

Change signs:
4x² + 3x – 5 – 2x² + x – 7

Combine:
2x² + 4x – 12

Multiplication of Polynomials

To multiply polynomials, use the distributive property. Multiply each term in the first expression by each term in the second.

Example:
Multiply (3x – 2) and (x + 4)

Solution:
3x(x + 4) – 2(x + 4)
= 3x² + 12x – 2x – 8
= 3x² + 10x – 8

Division of Polynomials

To divide polynomials, use long division if needed. The result might not always be a polynomial.

Example:
Divide 6x³ + 5x² – x – 3 by x + 2

Use long division to get:
6x² – 7x + 13 with a remainder

Because of the remainder, the result isn’t a pure polynomial.

Factoring Polynomials

Factoring rewrites a polynomial as a product of simpler expressions. This is useful for solving equations.

Example:

x² + 7x + 12

To factor this polynomial, find two numbers that multiply to 12 and add to 7. Those numbers are 3 and 4.

So, x² + 7x + 12 = (x + 3)(x + 4)

As you’ll see in the next section, factoring polynomials is a key part of solving them.

Solving Polynomials

To solve a polynomial means to find the values of the variable (usually x) that make the whole expression equal to zero. These values are called the solutions or roots of the polynomial.

In other words, you’re finding the inputs that make the output of the expression equal to zero. If you plug a solution into it, the result is zero.

For example, if you solve the equation x² – 4 = 0 and find that x = 2 and x = -2, that means:

• P(2) = 2² – 4 = 0
• P(-2) = (-2)² – 4 = 0

Each solution is a root of the polynomial, and each root corresponds to an x-intercept if you graph the function.

There are different strategies for solving polynomials depending on the degree.

Linear Example

4x – 8 = 0
Add 8: 4x = 8
Divide: x = 2
This equation has one root: x = 2

Quadratic Equation Example

x² – 3x – 10 = 0
Factor: (x – 5)(x + 2) = 0
Set each factor equal to 0:
x – 5 = 0 → x = 5
x + 2 = 0 → x = -2
This equation has two roots: x = 5 and x = -2.

If a quadratic polynomial doesn’t factor easily, you can use the quadratic formula to solve it. Learn how in my post about solving quadratic equations.

Higher-degree polynomials

For expressions of degree 3 or higher, solving might involve:

• Factoring (if possible)
• Using the Rational Root Theorem
• Graphing or using a table of values
• Synthetic or long division
• Numerical methods or a graphing calculator

Not all of these expressions can be factored easily. Some can have complex roots (with imaginary numbers), especially if they don’t cross the x-axis.

Finding the roots of a polynomial helps you understand its graph, solve real-world problems, and simplify expressions.

Advanced FAQ: Polynomial Properties and Theorems

What is the Remainder Theorem?

The Remainder Theorem helps you find the remainder when a polynomial is divided by a binomial like (x – a). Instead of doing full division, you can just plug in the value of a into the expression. The result is the remainder.

Example:

Say you want to divide P(x) = x² + 2x + 3 by (x – 1).

Instead of dividing, plug 1 into P(x):

P(1) = (1)² + 2(1) + 3 = 6

So, the remainder is 6.

What is the Factor Theorem?

The Factor Theorem says that if plugging a value into a polynomial gives you zero, then (x – that value) is a factor of that expression.

Example:

If P(2) = 0, then (x – 2) is a factor of P(x).

You can then factor it out just like you do with numbers.

What is the Division Algorithm?

This rule says that if you divide one polynomial by another, you can write it like this:

P(x) = (Divisor) × (Quotient) + Remainder

Just like when you divide numbers.

Example:

20 ÷ 6 = 3 with remainder 2

So, 20 = 6 × 3 + 2

Same idea works with polynomials.

What is Descartes’ Rule of Signs?

Descartes’ Rule of Signs helps you predict how many positive or negative real roots a polynomial might have. It doesn’t give you exact answers, but it narrows down the possibilities.

Step 1: Count sign changes in P(x) for positive real roots

Look at the signs of the coefficients in the original polynomial, written in standard form (highest degree to constant). Every time the sign switches from + to – or – to +, that’s a sign change.

Each sign change means at most one positive real root. The actual number of positive real roots is either equal to the number of sign changes or less by an even number (subtract 2, 4, etc.).

Step 2: Use P(-x) to find possible negative real roots

Now substitute –x in place of x in the polynomial. Simplify the expression, and then count the sign changes in P(-x).

Each sign change in P(-x) gives you the maximum number of negative real roots. Again, subtract by even numbers to get other possibilities.

Example

Let’s use this function:

P(x) = x^3 – 2x^2 + 5x – 8

Positive real roots (look at P(x))

Signs: +, –, +, –

There are 3 sign changes, so the number of positive real roots could be:

• 3 (maximum)
• or 1 (3 – 2)

Negative real roots (now use P(-x))

First, substitute –x into P(x):

P(-x) = (–x)³ – 2(–x)² + 5(–x) – 8

= –x³ – 2x² – 5x – 8

Signs: –, –, –, – → no sign changes

That means there are 0 negative real roots.

So in this example:

• There are either 3 or 1 positive real roots
• There are 0 negative real roots

The remaining roots (if any) must be complex (involving imaginary numbers).

What is the Fundamental Theorem of Algebra?

This theorem says that every polynomial with a degree of 1 or more has at least one complex root. That means you can always find a solution, even if it’s an imaginary number like 2 + 3i.

Why do complex roots come in pairs?

If the expression has real coefficients and one of its roots is a complex number like 4 – i, then its opposite, 4 + i, is also a root. These roots always show up in conjugate pairs.

FAQ About Polynomials

What is a polynomial in math?

It is an expression made of variables, constants, and exponents that are whole numbers and combined using plus and minus signs.

What is the degree of a polynomial?

It’s the highest exponent in the expression.

What does a polynomial function mean?

It’s a function where the rule is a polynomial.