Question

How do you use the rational roots theorem to find all possible zeros of `f(x)=3x^3-x^2+3x-1`?

Answer 1

The possible zeros are `1, -1, 1/3, -1/3`.

Explanation:

Find all the possible zeros of `f(x)=color(blue)3x^3-x^2+3xcolor(red)(-1)`.

The rational roots (or rational zeros) theorem states that if the polynomial has integer coefficients, the POSSIBLE roots(zeros) are the factors of the constant term divided by the factors of the leading coefficient.

Sometimes the factors of the constant term are referred to as "p" and the factors of the leading coefficient are referred to as "q". Then the possible zeros are given by `p/q`.

In this example, the constant term `=color(red)(-1)` and the factors are

`p=+1`

The leading coefficient `=color(blue)3` and the factors are

`q=+-1, +-3`

`p/q=frac(+-1)(+-1,+-3)= 1, -1, 1/3, -1/3`