Question

how do you find the zeros of the function f(x)=x(x-1)(x-5)^3

Answer 1

you set that mess equal to zero:
x(x-1)(x-5)^3 =0

now use this idea: if any of the 3 terms (x or (x-1) or (x-5)^3 is zero) then the whole thing is zero. Think about that.

That means you can break this into 3 separate problems:
x= 0
x-1 = 0
(x-5)^3 = 0

Answer 2

thank you, that was VERY helpful!

Answer 3

I should add that
(x-5)^3 = 0
can be written as
(x-5)(x-5)(x-5) =0
which turns into 3 (identical) problems:
x-5 =0
x-5= 0
x-5 =0

so all to0gther there are 5 separate equations
x= 0
x-1 = 0
x-5 =0
x-5= 0
x-5 =0

and 5 solutions: 0, 1 , 5, 5, 5
the 5's are called "repeated roots"