Question

In the equation f(x) = (3x + 5) (x2 - 6x + 9)2 = 0, 3 has a multiplicity of ______.

Answer 1

Multiplicity is how many times the root (or zero) is repeated

example, in $f(x)=a(x-b)^n(x-d)^m...$ 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 $f(x)=a(x-b)^n(x-d)^m$
for (3x+5), we need to force factor out the 3 to get
$3(x+\frac{5/3})$

factor the other one
$(x^2-6x+9)=(x-3)(x-3)=(x-3)^2$
but it is $(x^2-6x+9)^2$ which is equivilent to $((x-3)^2)^2$ which simplifies to $(x-3)^4$

so we get
$f(x)=3(x+\frac{5}{3})(x-3)^4$
so the roots are -5/3 multiplicity 1 and 3 multiplicity 4


3 has a multilicity of 4