Question

What are all the zeroes of `f(x) = 2x^3 - 2x^2 - 8x + 8`?

Answer 1

`x=-2,1,2`

Explanation:

Factor the polynomial.

`f(x)=2(x^3-x^2-4x+4)`

Factor by grouping.

`f(x)=2(x^2(x-1)-4(x-1))`

`f(x)=2(x^2-4)(x-1)`

Recognize that `x^2-4` is a difference of squares.

`f(x)=2(x+2)(x-2)(x-1)`

Now, to find the zeros of a function, find the times when `f(x)=0`.

`0=2(x+2)(x-2)(x-1)`

When there is a product of things that equal `0`, at least one of the terms MUST equal `0`. To find when this happens, set each non-constant term equal to `0`.

`x+2=0`
`color(blue)(x=-2)`

`x-2=0`
`color(blue)(x=2)`

`x-1=0`
`color(blue)(x=1)`

A good way to double check is by checking a graph.
graph{2x^3-2x^2-8x+8 [-7.96, 12.04, -3.56, 6.44]}

The zeros, when the graph crosses the `x`-axis, occur at `x=-2,1,2`.