Question

form a polynomial function f(x) with real coefficients having the given degree and zeros. Answers will vary depending on the choice of the leading coefficient. Use a graphing utility to graph the function and verify the result.

Degree 4; zeros: i, 1 + 2i;

Short answer

The polynomial can be given as$f\left(x\right)=x^4-2x^3+6x^2-2x+5$

Step-by-step solution

Given information

We are given the degree of polynomial to be 4 the zeros are 1+2i and i

Find the equation

As i and 1+2i are the root by conjugate property -i and 1-2i are also the roots of the polynomial

Hence

$$\begin{gathered} f\left(x\right)=(x+i)(x-i)(x-(1+2i\left)\right)(x-(1-2i\left)\right) \\ f\left(x\right)=(x^2+1)\left(\right(x-1)+2i)\left(\right(x-1)-2i) \\ f\left(x\right)=(x^2+1)({(x-1)}^2+4)f\left(x\right)=(x^2+1)(x^2-2x+5) \\ f\left(x\right)=x^4-2x^3+6x^2-2x+5 \end{gathered}$$

Conclusion

The polynomial satisfying the given condition is

$f\left(x\right)=x^4-2x^3+6x^2-2x+5$