Question

find the complex zeros of each polynomial function. Write f in factored form.

$f\left(x\right)=x^3-1$

Short answer

The complex root of the polynomial are$\frac{-1+3i}{2},\frac{-1-3i}{2}$.

Step-by-step solution

Given information

We are given a polynomial$f\left(x\right)=x^3-1$

Find the zeros of the polynomial

We get,

$$\begin{gathered} x^3-1=0 \\ (x-1)(x^2+x+1)=0 \end{gathered}$$

Now we use quadratic formula to find the remaining factors

$$\begin{gathered} x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ x=\frac{-1\pm \sqrt{1-4}}{2} \\ x=\frac{-1\pm \sqrt{-3}}{2} \\ x=\frac{-1\pm 3i}{2} \end{gathered}$$

Conclusion

The complex roots of the polynomial are$\frac{-1+3i}{2},\frac{-1-3i}{2}$