Question

How do you find all zeros of `f(x)=x^3-4x^2-25x+100`?

Answer 1

`x=5`, `x=-5` or `x=4`

Explanation:

Note that the ratio of the first and second terms is the same as that of the third and fourth terms, so this cubic factors by grouping and we find:

`0 = x^3-4x^2-25x+100`

`color(white)(0) = (x^3-4x^2)-(25x-100)`

`color(white)(0) = x^2(x-4)-25(x-4)`

`color(white)(0) = (x^2-25)(x-4)`

`color(white)(0) = (x^2-5^2)(x-4)`

`color(white)(0) = (x-5)(x+5)(x-4)`

Hence `x=5`, `x=-5` or `x=4`