Find the zeros of the polynomial function and state the multiplicity of each.
-$$f ( x ) = 4 ( x + 2 ) ^ { 3 } ( x - 4 ) ( x - 3 + i )$$
A) 2 (multiplicity 3), $- 4$ (multiplicity 1), $- 3 + \mathrm { i }$ (multiplicity 1)
B) 2 (multiplicity 3), 4 (multiplicity 1), 3 - i (multiplicity 1)
C) $- 2$ (multiplicity 3), $- 4$ (multiplicity 1), 3 - i (multiplicity 1)
D) -2 (multiplicity 3), 4 (multiplicity 1 ), 3 - i (multiplicity 1 )

Question

Quizplus · Accepted Answer

The zeros of the polynomial are found by setting each factor equal to zero. The factor $(x + 2)^3$ gives a zero at $x = -2$ with multiplicity 3 because the factor is raised to the third power. The factor $(x - 4)$ gives a zero at $x = 4$ with multiplicity 1. The factor $(x - 3 + i)$ is a complex zero, and since complex zeros occur in conjugate pairs in polynomials with real coefficients, the correct listing should include its conjugate, but since it's not explicitly given, we only consider the given zero, $3 - i$, with multiplicity 1.

Find the Zeros of the Polynomial Function and State the Multiplicity