Leave any unneeded answer spaces blank.
[Hint: use synthetic division to solve.]
f(x) = (1x^3 - 3x^2 - 25x + 75)/(1x^3 - 19x^2 + 118x - 240)
the roots of f(x), in increasing order is/are.... ___, ___, ___
f(x) has holes when x is... ___, ___, ___
f(x)has vertical asymptotes when x is....___, ___, ___
f(x) has a horizontal asymptote at y = ___
by inspection, the numerator is (x^2-25)(x-3)
The denominator is (x-5)(x-6)(x-8)
so, there is a hole at x=5, but zeros at x = -5 and x=3
vertical asymptotes at x=6,8
hor asymp at y=1
To find the roots of a function using synthetic division, we need to perform the following steps:
Step 1: Set up the synthetic division.
Given the function:
f(x) = (1x^3 - 3x^2 - 25x + 75)/(1x^3 - 19x^2 + 118x - 240)
We will set up the synthetic division using the coefficients of the numerator and denominator polynomials:
3 | 1 -3 -25 75
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Step 2: Perform the synthetic division.
Following synthetic division, we divide the coefficients of the numerator polynomial by the leading term of the denominator polynomial:
3 | 1 -3 -25 75
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Bring down the first coefficient of the numerator:
3 | 1 -3 -25 75
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1
Multiply the divisor (3) by the result (1) and add it to the next coefficient of the numerator:
3 | 1 -3 -25 75
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1
3
Repeat the process until all coefficients are used:
3 | 1 -3 -25 75
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1
3 0
9
27
Step 3: Analyze the results.
After performing synthetic division, we obtain the resulting polynomial:
1x^2 + 0x + 9 + 27/(3x + 0)
Simplifying the polynomial, we get:
1x^2 + 9/(3x)
Now, we can determine the roots, holes, and asymptotes based on the simplified polynomial:
The roots of f(x), in increasing order, are: No roots found.
f(x) has holes when x is: -3
f(x) has vertical asymptotes when x is: 0
f(x) has a horizontal asymptote at y = 1