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 5; zeros:2,-i, 1 + i;

Short answer

The polynomial satisfying the given conditions can be given as$f\left(x\right)=x^5-4x^4+7x^3-8x^2+6x-4$

Step-by-step solution

Given information

We are given the degree of polynomial to be 5 the zeros are 1+i , -i and 2

Find the equation

As 1+i and -i are the root of the polynomial hence by conjugate properties 1-i and i are also the root of the polynomial

Hence

$$\begin{gathered} f\left(x\right)=(x-i)(x+i)(x-2)(x-(1+i\left)\right)(x-(1-i\left)\right) \\ f\left(x\right)=(x-2)(x^2+1)\left(\right(x-1)+i)\left(\right(x-1)-i) \\ f\left(x\right)=(x-2)(x^2+1)({(x-1)}^2+1) \\ f\left(x\right)=(x-2)(x^2+1)(x^2-2x+2) \\ f\left(x\right)=x^5-4x^4+7x^3-8x^2+6x-4 \end{gathered}$$

Conclusion

The polynomial satisfying the given conditions can be given as

$f\left(x\right)=x^5-4x^4+7x^3-8x^2+6x-4$