Question

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

$h\left(x\right)=3x^4+5x^3+25x^2+45x-18;zero:3i$

Short answer

The roots of the polynomial are$3i,-3i,-2,\frac{1}{3}$

Step-by-step solution

Given information

We are given a polynomial$h\left(x\right)=3x^4+5x^3+25x^2+45x-1$

Find the roots using synthetic division

As 3i is one the roots of the polynomial by conjugate pairs property -3i is also a root of the polynomial

We use synthetic division to find the remaining roots of the polynomial

Hence,

$3i352545-189i15i-27-45i-6i1835+9i15i-2-6i0$

Now we use -3i in the synthetic division

role="math" localid="1647482204687" $-3i35+9i15i-2-6i-9i-15i6i35-20$

Hence we have,

$h\left(x\right)=(x-3i)(x+3i)(3x^2+5x-2)$

Use factorization to find the remaining roots

$$\begin{gathered} 3x^2+5x-2=0 \\ (3x-1)(x+2)=0 \\ x=\frac{1}{3}orx=-2 \end{gathered}$$

Conclusion

The roots of the polynomial are$3i,-3i,\frac{1}{3},-2$