Multiple Root — Definition, Formula & Examples

A multiple root is a root of a polynomial that occurs more than once, meaning the corresponding factor appears repeated in the polynomial's factored form. For example, in

(x - 3)^2 = 0

, the root

x = 3

is a multiple root because the factor

(x - 3)

appears twice.

A root $r$ of a polynomial $p(x)$ is called a multiple root (or repeated root) of multiplicity $k$ if $(x - r)^k$ divides $p(x)$ but $(x - r)^{k+1}$ does not, where $k \geq 2$ . When $k = 1$ , the root is called a simple root.

How It Works

To find whether a root is a multiple root, factor the polynomial completely and examine the exponent on each factor. If a factor

(x - r)

is raised to a power

k \geq 2

, then

r

is a multiple root with multiplicity

k

. Graphically, a root of even multiplicity causes the curve to touch the

x

-axis and bounce back, while a root of odd multiplicity causes the curve to cross the axis but with a flattened shape. The multiplicity of every root also affects derivative behavior: at a root of multiplicity

k

, the first

k - 1

derivatives of the polynomial are also zero.

Worked Example

Problem: Find all roots of

p(x) = x^3 - 5x^2 + 8x - 4

and identify any multiple roots.

Step 1: Test

x = 1

: substitute into the polynomial.

p(1) = 1 - 5 + 8 - 4 = 0

Step 2: Divide

p(x)

(x - 1)

using synthetic or polynomial division.

x^3 - 5x^2 + 8x - 4 = (x - 1)(x^2 - 4x + 4)

Step 3: Factor the quadratic.

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

Step 4: Write the full factorization and identify multiplicities.

p(x) = (x - 1)(x - 2)^2

Answer: The roots are

x = 1

(simple root, multiplicity 1) and

x = 2

(multiple root, multiplicity 2).

Why It Matters

Multiple roots appear frequently in optimization problems in calculus, where a function's derivative has a repeated root at a critical point. In engineering and physics, repeated roots of characteristic equations determine whether a system's response involves polynomial-times-exponential terms rather than pure exponentials. Recognizing multiplicity also helps you sketch polynomial graphs quickly and accurately.

Common Mistakes

Mistake: Counting a multiple root as separate distinct roots when listing all roots of a polynomial.

Correction: A root of multiplicity

k

is still one distinct root — it simply appears

k

times in the factorization. A cubic like

(x-2)^2(x+1)

has two distinct roots, not three, even though the degree is 3.