Question

Write a function that meets the given conditions.

Fourth-degree polynomial function with real coefficients; zeros:$-2,0,3+i$

Short answer

The polynomial function is$f\left(x\right)=x^4-4x^3-2x^2+20x$

Step-by-step solution

Step 1. Given Information

The given conditions are fourth-degree polynomial function with real coefficients; zeros:$-2,0,3+i$

Explanation

Since, the polynomial has a degree 4, the polynomial will have 4 zeroes.

Thus, $3-i$must also be a zero of the function.

Use the factor theorem and distributive property to find the polynomial.

localid="1646309910774" $$\begin{gathered} f\left(x\right)=a(x+2)(x+0)[x-(3+i\left)\right][x-(3-i\left)\right] \\ =ax(x+2)[x^2-3x+xi-3x-xi+9-i^2] \\ =a(x^2+2x)\left[x^2-6x+10\right] \\ =a(x^4-6x^3+10x^2+2x^3-12x^2+20x) \\ =a(x^4-4x^3-2x^2+20x) \end{gathered}$$

Let$a=1$then the required polynomial is$x^4-4x^3-2x^2+20x$