Question

Information is given about a complex polynomial function f(x) whose coefficients are real numbers. Find the remaining zeros of f. Then find a polynomial function with real coefficients that has the zeros.

Degree 4, Zeroes:$i,1+i$

Short answer

The remaining zeroes are $-i,1-i$and the polynomial is$x^4-2x^3+3x^2-2x+2$

Step-by-step solution

Step 1. Given Information

The given data is Degree 4, Zeroes:$i,1+i$

Step 2. Explanation

For a complex polynomial function with real coefficients, if zero is a complex number then its conjugate will also be a zero of the function.

Since $i,1+i$ is complex, their conjugate will be the remaining zero.

$$\begin{gathered} \overline{i}=-i \\ \overline{1+i}=1-i \end{gathered}$$

Step 3. Calculation 

Use the distributive property to find the polynomial using the factors.

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