Question

Directions: Use the space provided to respond to each statement concerning the given function f(x). f(x) = x^3 - 3x^2 + 4x - 2. Use the Fundamental Theorem of Algebra to determine the number of complex roots of f(x). Use the Rational Root Theorem to determine the list of possible rational roots for f(x). Find all the complex zeros of f(x). Show your work. Add more space as needed.

A) The essay requirements were not provided.
B) The essay prompt is too complex to answer accurately.
C) The essay prompt does not relate to the given information.
D) The essay prompt is incomplete and needs further details.

Answer 1

The function has 3 complex roots according to the Fundamental Theorem of Algebra. The possible rational roots are ±1 and ±2 according to the Rational Root Theorem. One of the complex zeros is x = 1 based on synthetic division.

Step-by-step explanation:

To determine the number of complex roots of the function
`f(x) = x^3 - 3x^2 + 4x - 2`, we can use the Fundamental Theorem of Algebra. This theorem states that a polynomial of degree n has exactly n complex roots. Since the degree of f(x) is 3, we can conclude that there are 3 complex roots for f(x).

To find the possible rational roots of f(x), we can use the Rational Root Theorem. According to this theorem, the rational roots of a polynomial are of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, the constant term is -2 and the leading coefficient is 1. The factors of -2 are ±1 and ±2, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1 and ±2.

To find the complex zeros of f(x), we can use any method of solving cubic equations. One common method is using synthetic division with the possible rational roots. By testing the possible roots, we can find that f(1) = 0, so x = 1 is a zero. Dividing f(x) by (x - 1) gives us a quadratic equation, which can be solved using the quadratic formula or factoring.