if $f(x)$ is a third degree polynomial function, how many distinct imaginary roots are possible?\n○ 0 or…

if $f(x)$ is a third degree polynomial function, how many distinct imaginary roots are possible?\n○ 0 or 2\n○ 0, 1, 2, or 3\n○ 1 or 2\n○ 1, 2, or 3

if $f(x)$ is a third degree polynomial function, how many distinct imaginary roots are possible?\n○ 0 or 2\n○ 0, 1, 2, or 3\n○ 1 or 2\n○ 1, 2, or 3

Answer

Explanation:

Step1: Recall conjugate root theorem

Imaginary roots of polynomials with real coefficients come in conjugate pairs.

Step2: Use degree of polynomial

A 3rd-degree polynomial has exactly 3 roots (counting multiplicities, Fundamental Theorem of Algebra).

Step3: Evaluate possible imaginary roots

Since imaginary roots are paired, we can have 0 (all real roots) or 2 (one real root, one pair of conjugate imaginary roots). We cannot have 1 or 3, as that would violate the conjugate root theorem.

Answer:

0 or 2