Polynomial Equation — Definition, Formula & Examples

A polynomial equation is an equation where a polynomial is set equal to zero (or to another polynomial). Solving it means finding every value of the variable that makes the equation true.

A polynomial equation in one variable $x$ is an equation that can be written in the form $P(x) = 0$ , where $P(x)$ is a polynomial of degree $n \geq 1$ with real or complex coefficients. Its solutions are called roots, or zeros of $P$ . By the Fundamental Theorem of Algebra, such an equation has exactly $n$ complex roots, counted with multiplicity.

Key Formula

a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 = 0, \quad a_n \neq 0

Where:

$x$ = The variable (unknown) you solve for
$a_n, \ldots, a_0$ = Coefficients (constants); $a_0$ is the constant term
$n$ = The degree, a positive integer; it is the maximum number of distinct roots

Worked Example

Problem: Solve the cubic equation

x^3 - x^2 - 4x + 4 = 0

Step 1: Group the first two terms and the last two terms. Then factor out the greatest common factor from each group.

x^2(x - 1) - 4(x - 1) = 0

Step 2: Factor out the common binomial

(x - 1)

. Then notice that

x^2 - 4

is a difference of squares.

(x - 1)(x^2 - 4) = (x - 1)(x - 2)(x + 2) = 0

Step 3: Apply the zero product property. A product equals zero only when at least one factor equals zero, so set each factor equal to zero.

x - 1 = 0, \quad x - 2 = 0, \quad x + 2 = 0

Answer: The solutions are

x = -2

x = 1

, and

x = 2

. The degree is 3 and there are 3 roots, which is the most a cubic can have.

Why It Matters

In Algebra 2 and Precalculus, polynomial equations are how you find the x-intercepts of a graph. Engineers solve them to find when a projectile hits the ground and to find the dimensions that give a box a target volume. In calculus, you solve the polynomial equation

f'(x) = 0

to locate the maximum and minimum points of a function.

Common Mistakes

Mistake: Using the zero product property when the right side is not zero. For example, a student turns

(x - 1)(x - 2) = 6

into

x - 1 = 6

x - 2 = 6

Correction: The property works only for zero. Expand and move everything to one side first:

x^2 - 3x - 4 = 0

, which factors as

(x - 4)(x + 1) = 0

. The solutions are

x = 4

and

x = -1

Related Terms

Equation of a Circle — the broader concept this term builds on
Characteristic Equation — the broader concept this term builds on
Equation — the broader concept this term builds on
Algebraic Equation — the broader concept this term builds on
Degree of a Polynomial — Sets the maximum number of roots
Cubic Polynomial — Degree-3 case, like the worked example
Conjugate Pair Theorem — Complex roots of real polynomials come in pairs
Coefficient in Math: Definition & Examples — The constants multiplying each power of x
Constant Polynomial — Degree-0 case, which gives no polynomial equation
Degree of a Term — Highest term degree determines the equation's degree