a) The polynomial f(x) in <em>expanded</em> form is f(x) = x³ + 10 · x² - 20 · x - 24.
b) The <em>rational</em> function g(x) in <em>factored</em> form is g(x) = [(x - 6) · (x + 3)] / (x - 2). there is no <em>slant</em> asymptotes.
c) There is one <em>evitable</em> discontinuity at x = - 1, and one <em>definitive </em>discontinuity at x = 2, where there is a <em>vertical</em> asymptote.
<h3>How to analyze polynomial and rational functions</h3>
a) In the first part of this question we need to determine the equation of a polynomial in <em>expanded</em> form, derived from its <em>factor</em> form defined below:
f(x) = Π (x - rₐ), for a ∈ {1, 2, 3, 4, ..., n} (1)
Where rₐ is the a-th root of the polynomial.
If we know that r₁ = 6, r₂ = - 1 and r₃ = - 3, then the polynomial in factor form is:
f(x) = (x - 6) · (x + 1) · (x + 3)
f(x) = (x - 6) · (x² + 4 · x + 4)
f(x) = (x - 6) · x² + (x - 6) · (4 · x) + (x - 6) · 4
f(x) = x³ - 6 · x² + 4 · x² - 24 · x + 4 · x - 24
f(x) = x³ + 10 · x² - 20 · x - 24
The polynomial f(x) in <em>expanded</em> form is f(x) = x³ + 10 · x² - 20 · x - 24.
b) The <em>rational</em> function is introduced below:
g(x) = (x³ + 10 · x² - 20 · x - 24) / (x² - x - 2)
g(x) = [(x - 6) · (x + 1) · (x + 3)] / [(x - 2) · (x + 1)]
g(x) = [(x - 6) · (x + 3)] / (x - 2)
The slope of the <em>slant</em> asymptote is:
m = lim [g(x) / x] for x → ± ∞
m = [(x - 6) · (x + 3)] / [x · (x - 2)]
m = 1
And the intercept of the <em>slant</em> asymptote is:
n = lim [g(x) - m · x] for x → ± ∞
n = Non-existent
Hence, there is no <em>slant</em> asymptotes.
c) There is <em>vertical</em> asymptote at a x-point if the denominator is equal to zero. There is one <em>evitable</em> discontinuity at x = - 1, and one <em>definitive </em>discontinuity at x = 2, where there is a <em>vertical</em> asymptote.
To learn more on asymptotes: brainly.com/question/4084552
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