Question

one root of a third degree polynomial function f (x) is -5 + 2i. which statement describes the number and nature of all roots for this function? f (x) has three imaginary roots. f (x) has two real roots and one imaginary root. f (x) has three real roots. f (x) has two imaginary roots and one real root.

Explanation:

Step1: Recall Complex Conjugate Root Theorem

For a polynomial with real coefficients, if \(a + bi\) is a root, then its complex conjugate \(a - bi\) is also a root. Here, the root is \(-5 + 2i\), so its conjugate \(-5 - 2i\) must also be a root.

Step2: Determine Number of Roots

A third - degree polynomial has 3 roots (by the Fundamental Theorem of Algebra). We already have two complex (imaginary) roots: \(-5 + 2i\) and \(-5 - 2i\). So the third root must be a real number (because the total number of roots is 3). So the polynomial has two imaginary roots and one real root.

Answer:

\(f(x)\) has two imaginary roots and one real root.