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: 3 + 2i; 4 ; multiplicity 2

Short answer

The polynomial can be given as$f\left(x\right)=x^4-14x^3+77x^2-200x+208$

Step-by-step solution

Given information

We are given the degree of polynomial to be 4 the zeros are 3+2i and 4 with multiplicity 2

Find the equation

As 3+2i is a root by conjugate pairs property 3-2i is also a root of the equation

Hence

$$\begin{gathered} f\left(x\right)=a(x-(3+2i\left)\right)(x-(3-2i\left)\right)(x-4)(x-4) \\ Assumwa=1 \\ f\left(x\right)=(x-(3+2i\left)\right)(x-(3-2i\left)\right)(x-4)(x-4) \\ Simplify \\ f\left(x\right)=\left(\right(x-3)+2i)\left(\right(x-3)-2i){(x-4)}^2 \\ f\left(x\right)=({(x-3)}^2+4)(x^2-8x+16) \\ f\left(x\right)=(x^2-6x+9+4)(x^2-8x+16) \\ f\left(x\right)=(x^2-6x+13)(x^2-8x+16) \\ f\left(x\right)=x^4-14x^3+77x^2-200x+208 \end{gathered}$$

Conclusion

The polynomial satisfying the given condition is

$f\left(x\right)=x^4-14x^3+77x^2-200x+208$