Question

which polynomial function has a leading coefficient of 3 and roots -4, i, and 2, all with multiplicity 1?
$f(x)=(x - 3)(x + 4)(x - i)(x - 2)$
$f(x)=3(x + 4)(x - i)(x - 2)$
$f(x)=3(x + 4)(x - i)(x + i)(x - 2)$
$f(x)=(x - 3)(x + 4)(x - i)(x + i)(x - 2)$

Explanation:

Step1: Recall the polynomial from roots

If a polynomial has roots \( r_1, r_2, \dots, r_n \) with multiplicity 1 and leading coefficient \( a \), then the polynomial is \( f(x)=a(x - r_1)(x - r_2)\cdots(x - r_n) \). Also, for complex roots, complex conjugates are roots too. The root \( i \) has a conjugate \( -i \), so we must include \( (x + i) \) as a factor.

Step2: Check leading coefficient and roots

The leading coefficient is 3, so \( a = 3 \). The roots are \( -4 \) (so factor \( (x + 4) \)), \( i \) (factor \( (x - i) \)), \( -i \) (factor \( (x + i) \)), and \( 2 \) (factor \( (x - 2) \)). Combining these, the polynomial is \( f(x)=3(x + 4)(x - i)(x + i)(x - 2) \).

Answer:

\( f(x) = 3(x + 4)(x - i)(x + i)(x - 2) \)