Question

use the given zero to find the remaining zeros of each function.

$f\left(x\right)=x^4-7x^3+14x^2-38x-60;zero:1+3i$

Short answer

The roots of the polynomial are$1+3i,1-3i,6,-1$

Step-by-step solution

Given information

We are given a polynomial$f\left(x\right)=x^4-7x^3+14x^2-38x-60$

Find the roots of the polynomial using synthetic division

We have 1+3i as one root of the polynomial hence by conjugate properties 1-3i is also a root of the polynomial

hence

$1+3i1-714-38-601-3i-15+15i44+18i601-6-3i-1+15i6+18i0$

Now we use 1-3i in synthetic divsion

$1+3i1-6-3i-1+15i6+18i1+3i-5-15i-6-18i1-5-60$

hence we get,

$f\left(x\right)=(x-(1+3i\left)\right(x-(1-3i)\left)\right(x^2-5x-6)$

Now we use factoring to find the remaining roots

$$\begin{gathered} x^2-5x-6=0 \\ (x-6)(x+1)=0 \\ x=6orx=-1 \end{gathered}$$

Conclusion

The roots of the polynomial are$1+3i,1-3i,6,-1$