Question

if \\(f(x)\\) is a third degree polynomial function, how many distinct imaginary roots are possible?

• 0 or 2
• 0, 1, 2, or 3
• 1 or 3
• 1, 2, or 3

Explanation:

⚡ Using what you learned: Understanding Polynomial Functions

Step 1: Analyze the degree of the polynomial

A third-degree polynomial function has exactly \(3\) complex roots (including real and imaginary roots), counted with multiplicity, according to the Fundamental Theorem of Algebra.

Step 2: Apply the Complex Conjugate Root Theorem

For polynomial functions with real coefficients, imaginary roots must always occur in conjugate pairs.

• If \(a + bi\) is a root, then its conjugate \(a - bi\) must also be a root.
• This means the number of imaginary roots must always be an even number (such as \(0\) or \(2\)).

Step 3: Determine the possible number of imaginary roots

Since the total number of roots is \(3\):

• If there are \(0\) imaginary roots, there are \(3\) real roots.
• If there are \(2\) imaginary roots, there is \(1\) real root.
• It is impossible to have \(1\) or \(3\) imaginary roots because they must occur in pairs.

Thus, the possible number of distinct imaginary roots is \(0\) or \(2\).

Answer:

0 or 2