Question

How do you find all the zeros of `f(x) = 2(x-5)(x+4)^2`?

Answer 1

The zeros (where`f(x)=0`) are located at `x=5` and `x=-4`.

Explanation:

To find the zeros, or x intercepts, set the equation equal to zero.
`f(x)=0` at the x intercepts.

`2(x-5)(x+4)^2=0`

`x-5=0` and `x+4=0`

`x=5` and `x=-4`

The zero at `x=5` has a multiplicity of one, because the exponent on the factor `(x-5)` is one. A zero with odd multiplicity indicates the graph crosses the x-axis at that point.

The zero at `x=-4` has a multiplicity of two, because the exponent on the factor `(x-4)^2` is two. A zero with even multiplicity indicates the graph just touches the x axis and then turns back in the same direction, i.e. the graph does not cross the x-axis.