Question

How do you find the zeroes for `f(x)=x^3(x-2)^2`?

Answer 1

`x=0,2`

Explanation:

A zero is any point where `f(x)=0`.

So, we have to find when `x^3(x-2)^2=0`.

Notice how there are terms being multiplied by one another, equaling `0`. The only way things can have a product of `0` is if one of the things itself IS `0`.

So, to solve this, we can split apart `x^3(x-2)^2`.

We can say that any of these are true:

`x^3=0`

or

`(x-2)^2=0`

Solve for both of these:

`x^3=0`
`x=0`

`(x-2)^2=0`
`x-2=0`
`x=2`

Therefore `x=0,2` because both instances will give us an answer of `0`.

Look at a graph:
graph{x^3(x-2)^2 [-3.96, 7.14, -1.19, 4.357]}
The two zeros are indeed located at `0` and `2`.