Multiplicity of a Root

The multiplicity of a root $ r $ of a polynomial $ P(x) $ is the positive integer $ n $ such that: $$ P(x) = (x - r)^n Q(x) $$ where $ Q(x) $ is a polynomial and $ Q(r) \neq 0 $.

In essence, the multiplicity of a root $ r $ indicates how many times the factor $ (x - r) $ divides the polynomial $ P(x) $.

A Concrete Example

Let’s consider the polynomial:

$$ x^3 - x^2 - 2x $$

We begin by factoring it:

$$ x \cdot (x^2 - x - 2) $$

The first root is clearly $ r_1 = 0 $, since the factor $ x $ evaluates to zero at $ x = 0 $.

The remaining roots are found by solving the quadratic $ x^2 - x - 2 $:

$$ r_{2,3} = \frac{1 \pm \sqrt{1 + 8}}{2} = \begin{cases} r_2 = \frac{1 - \sqrt{9}}{2} = -1 \\ \\ r_3 = \frac{1 + \sqrt{9}}{2} = 2 \end{cases} $$

Now let's examine the root $ r = 2 $. To factor the quadratic $ x^2 - x - 2 $, we apply Ruffini’s synthetic division:

$$ \begin{array}{c|lcc|r} & 1 & -1 & -2 \\ 2 & & 2 & 2 \\ \hline & 1 & 1 & 0 \end{array} $$

This confirms the factorization:

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

Substituting back, the full factorization of the original polynomial becomes:

$$ x \cdot (x - 2)(x + 1) $$

This shows that the factor $ (x - 2) $ appears exactly once in the factorization, meaning it has exponent $ n = 1 $.

Therefore, the root $ r = 2 $ has multiplicity one in the original polynomial.

And so on.