To form a polynomial $f(x)$ with real coefficients, we need to follow these steps, given the information in the image:

1. Degree: The polynomial has degree 4.
2. Zeros and Multiplicities:
   • There is a real zero at $x = 3$ with multiplicity 2.
   • There is a complex zero at $x = 4i$ (and by the nature of polynomials with real coefficients, its conjugate $x = -4i$ must also be a zero).

Steps to Form the Polynomial

1. Real Zero with Multiplicity 2: Since $x = 3$ has a multiplicity of 2, it contributes the factor $(x - 3)^2$.

2. Complex Conjugate Zeros: Since $x = 4i$ and $x = -4i$ are zeros, they contribute the factors $(x - 4i)$ and $(x + 4i)$. Combining these, we get: $(x - 4i)(x + 4i) = x^2 - (4i)^2 = x^2 + 16$

3. Form the Polynomial: Combining all the factors, the polynomial $f(x)$ can be written as: $f(x) = a(x - 3)^2(x^2 + 16)$ where $a$ is the leading coefficient.

Final Answer:

$f(x) = a(x - 3)^2(x^2 + 16)$

Would you like a step-by-step expansion or simplification of this polynomial?

Related Questions

1. How do complex conjugates affect the structure of polynomials with real coefficients?
2. Why do complex roots always appear in conjugate pairs for polynomials with real coefficients?
3. How can the degree of a polynomial determine the number of roots (including multiplicities)?
4. How would changing the multiplicity of a root affect the polynomial's shape?
5. What happens if we include additional real or complex roots in a degree-four polynomial?

Tip: When dealing with polynomials, always check if the complex roots have their conjugates included to ensure all coefficients remain real.