Question

How do you find all zeros of the function `f(x)=x^3+3x^2-34x+48`?

Answer 1

The zeros are `2, 3,` and `-8`.

Explanation:

A simple way to find the zeros of this function is to factor the polynomial. There are several ways to find these factors, but for this question, let's take a computer-based short-cut! We can look for the zeros by plotting the function, and if we are lucky we'll see the integer values that correspond to the zeros:

graph{x^3+3x^2-34x+48 [-10, 10, -20, 20]}

from this we can guess that the zeros are `2, 3,` and `-8` which makes our function:

`f(x) = (x-2)(x-3)(x+8)`

if we multiply this out and regain our original cubic polynomial then our guesses are correct:

`f(x) = (x^2-5x+6)(x+8) = x^3+3x^2-34x+48`

therefore, our guesses are correct!