Question

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

$f\left(x\right)=x^3+13x^2+57x+85$

Short answer

The complex roots of the polynomial are 4+i and 4-i

Step-by-step solution

Given information

We are given a polynomial$f\left(x\right)=x^3+13x^2+57x+85$

Find the factors

To find the factors of the polynomial we first have to find the factors of 85,

They can be given as$\pm 1,\pm 5,\pm 17,\pm 85$

Let us try -5 using synthetic division

role="math" localid="1647572985088" $-51135785-5-40-8518170$

hence the polynomial becomes

role="math" localid="1647573001820" $$\begin{gathered} f\left(x\right)=(x+5)(x^2+8x+17) \\ Nowweusequadraticformulatofind \\ theremainingfactors \\ x=\frac{-8\pm \sqrt{64-68}}{2} \\ x=\frac{-8\pm 2i}{2} \\ x=4\pm i \end{gathered}$$

Conclusion

The complex roots of the polynomial are 4+i and 4-i