To determine the polynomial function that has a leading coefficient of 3 and roots [tex]\(-4\)[/tex], [tex]\(i\)[/tex], and [tex]\(2\)[/tex], all with multiplicity 1, we need to follow a step-by-step process.
1. Identify the Roots:
The given roots of the polynomial are:
[tex]\[
-4, \quad i, \quad 2
\][/tex]
2. Recognize the Conjugate Root Theorem:
When dealing with polynomials with real coefficients, if a complex number [tex]\(i\)[/tex] is a root, its complex conjugate [tex]\(-i\)[/tex] must also be a root. Therefore, we have:
[tex]\[
\text{Roots: } -4, \quad i, \quad -i, \quad 2
\][/tex]
3. Form the Factors:
Each root [tex]\(r\)[/tex] corresponds to a factor of the form [tex]\( (x - r) \)[/tex]. Thus, the polynomial will have the factors:
[tex]\[
(x + 4), \quad (x - i), \quad (x + i), \quad (x - 2)
\][/tex]
4. Write the Polynomial with the Given Leading Coefficient:
Since the leading coefficient is 3, we multiply the product of these factors by 3. The polynomial function becomes:
[tex]\[
f(x) = 3(x + 4)(x - i)(x + i)(x - 2)
\][/tex]
5. Simplify the Factors involving Complex Numbers:
Notice that:
[tex]\[
(x - i)(x + i) = x^2 - i^2 = x^2 + 1
\][/tex]
Thus, the polynomial can be written as:
[tex]\[
f(x) = 3(x + 4)(x^2 + 1)(x - 2)
\][/tex]
6. Verify the Correct Form:
We now check the options provided to find the one that matches our polynomial:
[tex]\[
\begin{aligned}
&A. \quad f(x) = 3(x+4)(x-1)(x-2) \quad \text{(Incorrect; contains an extraneous root)}\\
&B. \quad f(x) = (x-3)(x+4)(x-1)(x-2) \quad \text{(Incorrect; wrong leading factor and extraneous roots)}\\
&C. \quad f(x)= (x-3)(x+4)(x-i)(x+i)(x-2) \quad \text{(Incorrect; wrong leading coefficient)}\\
&D. \quad f(x) = 3(x+4)(x-i)(x+i)(x-2) \quad \text{(Correct)}
\end{aligned}
\][/tex]
Therefore, the correct polynomial function is:
[tex]\[
f(x) = 3(x+4)(x-i)(x+i)(x-2)
\][/tex]
Hence, the answer is:
[tex]\[
f(x) = 3(x+4)(x-i)(x+i)(x-2)
\][/tex]
