Multiplicity of Roots of Polynomials

A root a of a polynomial f (x) has multiplicity k if f (x) = (x - a)kg(x), where g(a)≠ 0. If k > 1, then f(x) = k(x - a)k-1g(x) + (x - a)kg(x) = (x - a)k-1(kg(x) + (x - a)g(x)), and hence gcd(f (x), f(x)) = (x - a)k-1h(x)≠1. This observation provides a test for multiple roots: If gcd$\left(\vphantom{ f(x),f^{\prime }(x)}\right.$f (x), f(x)$\left.\vphantom{ f(x),f^{\prime }(x)}\right)$ is a constant, then f (x) has no multiple roots; otherwise, f (x) has at least one multiple root.

The graphs of

f (x) = 5537x5 -34804x4 +60229x3 -29267x2 + 19888x + 54692

and

g(x) = 5537x5 -34797x4 +60207x3 -29260x2 + 19873x + 54670

appear indistinguishable. Both appear to have a root near 3.1.

itbpFU2.1577in1.7573in0inf (x) itbpFU2.1577in1.7573in0ing(x)

However,


$\blacktriangleright$ Evaluate

gcd(f (x), f(x)) =  791x - 2486


whereas


$\blacktriangleright$ Evaluate

gcd(g(x), g(x)) =  7


Thus, x = 2486/791 = 22/7 is a root of f (x) of multiplicity at least 2, whereas g(x) has no multiple roots. Solving f (x) = 0 and g(x) = 0, the real solutions are as follows.


$\blacktriangleright$ Solve + Exact

f (x) = 0, Solution is : $\left\{\vphantom{ x=\frac{22}{7}}\right.$x = ${\frac{{22}}{{7}}}$$\left.\vphantom{ x=\frac{22}{7}}\right\}$$\left\{\vphantom{ x=\frac{22}{7}}\right.$x = ${\frac{{22}}{{7}}}$$\left.\vphantom{ x=\frac{22}{7}}\right\}$,

         $\left\{\vphantom{ x=-\sqrt[3]{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right...
...frac{1}{3\sqrt[3]{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right) }}}\right.$x = - $\sqrt[3]{{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right) }}$ + ${\dfrac{{1}}{{3\sqrt[3]{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right) }}}}$$\left.\vphantom{ x=-\sqrt[3]{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right)...
...rac{1}{3\sqrt[3]{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right) }}}\right\}$

g(x) = 0, Solution is : $\left\{\vphantom{ x=\frac{22}{7}}\right.$x = ${\frac{{22}}{{7}}}$$\left.\vphantom{ x=\frac{22}{7}}\right\}$$\left\{\vphantom{ x=\frac{%
355}{113}}\right.$x = ${\frac{{%
355}}{{113}}}$$\left.\vphantom{ x=\frac{%
355}{113}}\right\}$,

         $\left\{\vphantom{ x=-\sqrt[3]{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right...
...frac{1}{3\sqrt[3]{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right) }}}\right.$x = - $\sqrt[3]{{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right) }}$ + ${\dfrac{{1}}{{3\sqrt[3]{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right) }}}}$$\left.\vphantom{ x=-\sqrt[3]{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right)...
...rac{1}{3\sqrt[3]{\left( \frac{1}{2}+\frac{1}{18}\sqrt{93}%
\right) }}}\right\}$


$\blacktriangleright$ Solve + Numeric

f (x) = 0, Solution is : $\left\{\vphantom{ x=-.68233}\right.$x = - .68233$\left.\vphantom{ x=-.68233}\right\}$$\left\{\vphantom{
x=3.1429}\right.$x = 3.1429$\left.\vphantom{
x=3.1429}\right\}$$\left\{\vphantom{
x=3.1429}\right.$x = 3.1429$\left.\vphantom{
x=3.1429}\right\}$

g(x) = 0, Solution is : $\left\{\vphantom{ x=-.68233}\right.$x = - .68233$\left.\vphantom{ x=-.68233}\right\}$$\left\{\vphantom{
x=3.1416}\right.$x = 3.1416$\left.\vphantom{
x=3.1416}\right\}$$\left\{\vphantom{
x=3.1429}\right.$x = 3.1429$\left.\vphantom{
x=3.1429}\right\}$


Thus, g has two distinct roots that are extremely close, whereas f has a root of multiplicity two.