Question

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

$f\left(x\right)=x^3-8x^2+25x-26$

Short answer

The complex roots of the polynomial are 3+2i, 3-2i

Step-by-step solution

Given information

We are given a polynomial$f\left(x\right)=x^3-8x^2+25x-26$

Find the factors

To find the factors of the polynomial we first find the factors of -26

They can be given as $\pm 1,\pm 2,\pm 13,\pm 26$

Let us test 2 using synthetic division whether we get zero or not

$21-825-262-12261-6130$

Hence the polynomial can be given as

$f\left(x\right)=(x-2)(x^2-6x+13)$

Now using quadratic formula find the remaining factors

$$\begin{gathered} x^2-6x+13=0 \\ x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ x=\frac{6\pm \sqrt{36-52}}{2} \\ x=\frac{6\pm 4i}{2} \\ x=3\pm 2i \end{gathered}$$

Conclusion

The complex roots of the polynomial are 3+2i, 3-2i