Finding the Zeros of a Polynomial Function with Complex Zeros

The Rational Zero Theorem tells us that if $\frac{p}{q}$ is a zero of $f(x),$ then $p$ is a factor of 3 and $q$ is a factor of 3.

The factors of 3 are $\mathrm{\pm 1}$ and $\mathrm{\pm 3.}$ The possible values for $\frac{p}{q},$ and therefore the possible rational zeros for the function, are $\mathrm{\pm 3},\text{±1, and }\pm \frac{1}{3}.$ We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Let’s begin with –3.

Dividing by $(x+3)$ gives a remainder of 0, so –3 is a zero of the function. The polynomial can be written as

We can then set the quadratic equal to 0 and solve to find the other zeros of the function.

The zeros of $f(x)$ are –3 and $\pm \frac{i\sqrt{3}}{3}.$