Question

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

$h\left(x\right)=x^4-9x^3+21x^2+21x-130;zero:3-2i$

Short answer

The roots of the polynomial are$3+2i,3-2i,5,-2$

Step-by-step solution

Given information

We are given a polynomial$h\left(x\right)=x^4-9x^3+21x^2+21x-130$

Find the roots using synthetic division

As 3+2i is a root of the polynomial by conjugate properties 3-2i is also a root of the polynomial

Hence

$3-2i1-92121-1303-2i-22+6i9+20i1301-6-2i-1+6i30+20i0$

Now we use 3+2i in synthetic division

$3+2i1-6-2i-1+6i30+20i3+2i-9-6i-30-20i1-3-100$

Hence we have,

$h\left(x\right)=(x-(3-2i\left)\right)(x-(3+2i\left)\right)(x^2-3x-10)$

Now we use facotring to find the remaining roots

$$\begin{gathered} x^2-3x-10=0 \\ (x-5)(x+2)=0 \\ x=5orx=-2 \end{gathered}$$

Conclusion

The roots of the polynomial are$3-2i,3+2i,5,-2$