multiplicity is how many times the root (or zero) is repeated
example, in [tex]f(x)=a(x-b)^n(x-d)^m...[/tex] the root b has a multiplicity of n and the root d has a multiplicity of m
first we need to get the roots into [tex]f(x)=a(x-b)^n(x-d)^m[/tex]
for (3x+5), we need to force factor out the 3 to get
[tex]3(x+\frac{5/3})[/tex]
factor the other one
[tex](x^2-6x+9)=(x-3)(x-3)=(x-3)^2[/tex]
but it is [tex](x^2-6x+9)^2[/tex] which is equivilent to [tex]((x-3)^2)^2[/tex] which simplifies to [tex](x-3)^4[/tex]
so we get
[tex]f(x)=3(x+\frac{5}{3})(x-3)^4[/tex]
so the roots are -5/3 multiplicity 1 and 3 multiplicity 4
3 has a multilicity of 4
