Quiz for Math 111 (CRN 13936), Week 7
Note: Show your work!
Name
1. f(x) = x3+ 7x2+ 4xโ12, g(x) = xโ1, is g(x) a factor of f(x)? Why? (20 points)
Solution:
Recall that โxโais a factor of polynomial f(x)โ just means โf(a) = 0โ. To judge
whether xโ1 is a factor of f(x), just plug 1 into f(x), we have f(1) = 13+7ยท12+4ยท1โ12 = 0,
so g(x) = xโ1 is a factor of f(x).
In this example, f(x)is relatively simple, you can also use polynomial divi-
sion to verify that g(x)is a factor of f(x)(using brute force)
2. Find the horizontal and vertical asymptotes algebraically for rational function f(x)
a) f(x) = x(xโ1)
(xโ2)3
b) f(x) = x(xโ4)
x(xโ5)
Is x= 0 a pole of function fin a)? Is x= 0 a pole of function fin b)? (30 points)
Solution:
Plug x= 0 into f(x) in a), the numerator is 0 and the denominator is not 0, that is
f(0) = 0
4= 0. So x= 0 is not a pole in a) (because 0 is in the domain of f(x) in a) ).
Plug x= 0 into f(x) in b), the numerator is 0 and the denominator is also 0, so x= 0
is not in the domain of f(x) in b), besides, for all the x6= 0, we can cancel xfrom the
numerator and denominator to get f(x) = xโ4
(xโ5) , x 6= 0. So x= 0 is a pole of f(x) in b).
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