Question

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

$f\left(x\right)=x^4-1$

Short answer

The complex roots of the polynomial are$i,-i$

Step-by-step solution

Given information

We are given a polynomial$f\left(x\right)=x^4-1$

Find the roots of the polynomial

We have,

$$\begin{gathered} x^4-1=0 \\ (x^2-1)(x^2+1)=0\left[\right(a^2-b^2)=(a-b\left)\right(a+b\left)\right] \\ (x-1)(x+1)(x^2+1)=0 \end{gathered}$$

Now to find complex roots

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

Conclusion

The complex roots of the polynomial are$i,-i$