What is the multiplicity of the real root of an equation that crosses/touches the x-axis once?

Note that #f(x) = x^3# has the properties:

• #f(x)# is of degree #3#

• The only real value of #x# for which #f(x) = 0# is #x=0#

Those two properties alone are not sufficient to determine that the zero at #x=0# is of multiplicity #3#.

For example, consider:

#g(x) = x^3+x = x(x^2+1)#

Note that:

• #g(x)# is of degree #3#

• The only real value of #x# for which #g(x) = 0# is #x=0#

But the multiplicity of the zero of #g(x)# at #x=0# is #1#.

Some things we can say:

• A polynomial of degree #n > 0# has exactly #n# complex (possibly real) zeros counting multiplicity. This is a consequence of the Fundamental Theorem of Algebra.

• #f(x) = 0# only when #x=0#, yet it is of degree #3#, so has #3# zeros counting multiplicity.

• Therefore that zero at #x=0# must be of multiplicity #3#.

Why is the same not true of #g(x)#?

It is of degree #3#, so has three zeros, but two of them are non-real complex zeros, name #+-i#.

Another way of looking at this is to observe that #x=a# is a zero of #f(x)# if and only if #(x-a)# is a factor.

We find:

#f(x) = x^3 = (x-0)(x-0)(x-0)#

That is: #x=0# is a zero #3# times over.