which polynomial function has a leading coefficient of 3 and roots -4, i, and 2, all with multiplicity 1?\n○…

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

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

Answer

Explanation:

Step1: Recall polynomial - root relationship

If (r) is a root of a polynomial, then ((x - r)) is a factor. Given roots (-4), (i), and (2) with multiplicity 1, the factors are ((x + 4)), ((x - i)), and ((x - 2)). For a polynomial with real - valued coefficients, if (i) is a root, then its complex conjugate (-i) is also a root, and the factor corresponding to the complex - conjugate root is ((x + i)).

Step2: Consider the leading coefficient

The general form of the polynomial with leading coefficient (a) and factors ((x - r_1)(x - r_2)\cdots(x - r_n)) is (f(x)=a(x - r_1)(x - r_2)\cdots(x - r_n)). Here, (a = 3), and the factors are ((x + 4)), ((x - i)), ((x + i)), and ((x - 2)). So the polynomial is (f(x)=3(x + 4)(x - i)(x + i)(x - 2)).

Answer:

D. (f(x)=3(x + 4)(x - i)(x + i)(x - 2))