Question

A polynomial f (x) has the given zeros of 8, –1, and –3.

Part A: Using the Factor Theorem, determine the polynomial f (x) in expanded form. Show all necessary calculations.

Part B: Divide the polynomial f (x) by (x2 – x – 2) to create a rational function g(x) in simplest factored form. Determine g(x) and find its slant asymptote.

Part C: List all locations and types of discontinuities of the function g(x). Be sure to check for all asymptotes and holes. Show all necessary calculations.

Answer 1

Part A:

f(x) = (x - 8)(x + 1)(x + 3)

= (x - 8)(x² + 4x + 3)

= x³ + 4x² + 3x - 8x² - 32x - 24

= x³ - 4x² - 29x - 24

Part B:

g(x) = (x³ - 4x² - 29x - 24)/(x² - x - 2)

= ((x - 8)(x + 1)(x + 3))/((x + 1)(x - 2))

= ((x - 8)(x + 3))/(x - 2)

The slant asymptote is y = x.

Part C:

Vertical asymptote of g(x): x = 2

Hole of g(x): x = -1

Slant asymptote of g(x): y = x