Question

What are all the possible rational zeros for `f(x)=x^3+3x^2-x-3` and how do you find all zeros?

Answer 1

All possible rational zeros are `-3`, `-1` and `1`.

Explanation:

If `a`, `b` and `c` are zeros of a polynomial of degree `3` in `x`, then the function is `(x-a)(x-b)(x-c)`. Hence let us factorize `x^3+3x^2-x-3`

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

= `x^2(x+3)-1(x+3)`

= `(x^2-1)(x+3)`

= `(x+1)(x-1)(x+3)`

Hence, all possible rational zeros are `-3`, `-1` and `1`.