Answer:
Part A:
Using the Factor Theorem, we know that f(x) can be written as:
f(x) = a(x - 6)(x + 1)(x + 3)
where a is a constant that we need to determine.
To find the value of a, we can use one of the given zeros of f(x), for example, x = 6. When x = 6, we know that f(x) = 0, so we can substitute these values into the equation above:
0 = a(6 - 6)(6 + 1)(6 + 3)
Simplifying, we get:
0 = 189a
Therefore, a = 0.
So, the polynomial f(x) is:
f(x) = 0(x - 6)(x + 1)(x + 3)
Simplifying, we get:
f(x) = 0
Part B:
To divide f(x) by (x² - x - 2), we can use long division:
0
___________
x² - x - 2 | 0x³ + 0x² + 0x + 0
- (0x³ - 0x² - 0x)
_______________
0x² + 0x
- (0x² - 0x - 2)
_______________
2x + 2
Therefore, the rational function g(x) in simplest factored form is:
g(x) = (2x + 2)/(x² - x - 2)
To find the vertical asymptotes, we need to find the roots of the denominator:
x² - x - 2 = 0
(x - 2)(x + 1) = 0
Therefore, the vertical asymptotes are x = 2 and x = -1.
To find the horizontal asymptote, we can use the fact that the degree of the numerator is less than the degree of the denominator. Therefore, the horizontal asymptote is y = 0.
Part C:
The function g(x) has two vertical asymptotes at x = 2 and x = -1.
To check for any holes, we can simplify the function by factoring the numerator:
g(x) = 2(x + 1)/(x - 2)(x + 1)
Since we factored the numerator and denominator of g(x), we can see that there is a hole in the graph at x = -1. This is because the factor (x + 1) cancels out in both the numerator and denominator, leaving a hole at that point.
To find the x-intercepts, we need to solve for when the numerator is equal to zero:
2x + 2 = 0
x = -1
Therefore, the x-intercept is (-1, 0).
To find the y-intercept, we can substitute x = 0 into the equation for g(x):
g(0) = 2(0 + 1)/(0 - 2)(0 + 1)
g(0) = -1
Therefore, the y-intercept is (0, -1).
To sketch the graph of g(x), we can use the information we have gathered so far. The graph has two vertical asymptotes at x = 2 and x = -1, a hole at x = -1, an x-intercept at (-1, 0), and a y-intercept at (0, -1). The horizontal asymptote is y = 0.
We can also use the factored form of g(x) to determine the end behavior of the graph. As x approaches positive or negative infinity, the function approaches zero. Therefore, the graph approaches the x-axis on either side of the vertical asymptotes.
Putting all of this information together, we can sketch the graph of g(x) as follows:
[insert graph of g(x) here]