To determine which polynomial function has a leading coefficient of 3 and roots [tex]\(-4\)[/tex], [tex]\(i\)[/tex], and [tex]\(2\)[/tex] (all with multiplicity 1), let's follow these steps:
1. Identify Factors from Roots:
- A root of [tex]\(-4\)[/tex] corresponds to a factor of [tex]\((x + 4)\)[/tex].
- A root of [tex]\(i\)[/tex] (imaginary unit) corresponds to factors of [tex]\((x - i)\)[/tex] and [tex]\((x + i)\)[/tex] because complex roots come in conjugate pairs.
- A root of [tex]\(2\)[/tex] corresponds to a factor of [tex]\((x - 2)\)[/tex].
2. Construct the Polynomial:
- The polynomial function can be constructed by multiplying these factors together: [tex]\( (x + 4)(x - i)(x + i)(x - 2) \)[/tex].
- Since the leading coefficient is given as 3, we multiply the entire polynomial by 3 to obtain: [tex]\( 3(x + 4)(x - i)(x + i)(x - 2) \)[/tex].
3. Match with Given Choices:
- Now, let's compare the constructed polynomial [tex]\(3(x + 4)(x - i)(x + i)(x - 2)\)[/tex] with the given choices:
- [tex]\( f(x) = 3(x + 4)(x - 1)(x - 2) \)[/tex]
- [tex]\( f(x) = (x - 3)(x + 4)(x - 1)(x - 2) \)[/tex]
- [tex]\( f(x) = (x - 3)(x + 4)(x - i)(x + i)(x - 2) \)[/tex]
- [tex]\( f(x) = 3(x + 4)(x - i)(x + i)(x - 2) \)[/tex]
From the comparison, the polynomial [tex]\( f(x) = 3(x + 4)(x - i)(x + i)(x - 2) \)[/tex] aligns with the polynomial we have constructed.
Therefore, the correct choice is:
[tex]\[ \boxed{4} \][/tex]
