Simple Root — Definition, Formula & Examples

A simple root is a root of a polynomial that occurs exactly once — meaning the corresponding linear factor appears only to the first power in the polynomial's factored form.

A value $r$ is a simple root of a polynomial $p(x)$ if $(x - r)$ is a factor of $p(x)$ but $(x - r)^2$ is not. Equivalently, $r$ is a root of multiplicity 1, so $p(r) = 0$ and $p'(r) \neq 0$ .

How It Works

To determine whether a root is simple, factor the polynomial completely and check the exponent on each linear factor. If a factor

(x - r)

appears with exponent 1, then

r

is a simple root. You can also use the derivative test: if

p(r) = 0

and

p'(r) \neq 0

, the root is simple. At a simple root, the graph of

p(x)

crosses the

x

-axis (rather than merely touching it and turning back), which makes simple roots visually distinct from repeated roots.

Worked Example

Problem: Determine which roots of

p(x) = (x - 1)(x + 3)^2

are simple roots.

Identify the roots: Set each factor equal to zero.

x - 1 = 0 \Rightarrow x = 1, \quad x + 3 = 0 \Rightarrow x = -3

Check multiplicities: The factor

(x - 1)

has exponent 1, so

x = 1

has multiplicity 1. The factor

(x + 3)

has exponent 2, so

x = -3

has multiplicity 2.

Verify with the derivative (optional): Compute

p'(x)

and evaluate at each root. Since

p'(1) = (1+3)^2 = 16 \neq 0

, the root

x = 1

is confirmed simple. Since

p'(-3) = 0

, the root

x = -3

is not simple.

p'(x) = (x+3)^2 + 2(x-1)(x+3)

Answer:

x = 1

is a simple root;

x = -3

is not (it is a repeated root of multiplicity 2).

Why It Matters

Distinguishing simple roots from repeated roots is essential when performing partial fraction decomposition in calculus and when analyzing the stability of systems in engineering. In numerical analysis, root-finding algorithms like Newton's method converge faster to simple roots than to repeated ones.

Common Mistakes

Mistake: Assuming every root that appears once in a list of solutions is simple.

Correction: A root's multiplicity depends on the exponent of its factor in the fully factored polynomial, not on how many times you happen to write it down. Always factor completely and check the exponent.