Question

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

$f\left(x\right)=2x^4+5x^3+5x^2+20x-12;zero:-2i$

Short answer

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

Step-by-step solution

Given information

We are given a polynomial$f\left(x\right)=2x^4+5x^3+5x^2+20x-12$

Find the zeros by synthetic divsion

We have,

$f\left(x\right)=2x^4+5x^3+5x^2+20x-12$and 2i as one of the root

$2i25520-124i10i-8-20-6i1225+4i10i-3-6i0$

as 2i is one root then -2i is the second root

$-2i25+4i10i-3-6i-4i-10i6i25-30$

Now we have,

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

Now we factor the remaining term

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

Conclusion

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