Question

How do you find all zeros with multiplicities of `f(x)=9x^3-5x^2-x`?

Answer 1

`x=0" "` or `" "x = 5/18+-sqrt(61)/18`

all with multiplicity `1`

Explanation:

`0 = 36f(x)`

`color(white)(0) = 36(9x^3-5x^2-x)`

`color(white)(0) = x(324x^2-180x-36)`

`color(white)(0) = x((18x)^2-2(18x)(5)+25-61)`

`color(white)(0) = x((18x-5)^2-(sqrt(61))^2)`

`color(white)(0) = x(18x-5-sqrt(61))(18x-5+sqrt(61))`

Hence:

`x=0`

or:

`18x=5+-sqrt(61)" "` so `" "x = 5/18+-sqrt(61)/18`