If the graph of the function y-(ax^2+b)/(x^2+cx+4) has vertical asymptotes x=1,x=4, a horizontal asymptote of y=2 and x-intercepts at x=+-2, find a, b and c.

Question

Answers: a=2, b=-8 and c=-5

I found a and b but I'm having trouble finding c.

I made y=0 and x=2
0=(4a+b)/(4+2c+4)
Then I simplified:
0=4a+b
-4a=b

I know that when the a term in the numerator and a term in the denominator both have the same highest exponent on the same variable then you take the coefficients of those two terms and divide them by each other. (You take the coefficient of that term in the numerator and divide it by the coefficient of that term in the denominator)

So the term in the numerator would be ax^2 with the coefficient of 'a' and the term in the denominator would be x^2 with the coefficient of '1'.
In the question, I'm given the information that the horizontal asymptote of this function is y=2, therefore I can conclude that a=2.

Referring back on the equation I previously formulated, -4a=b.
Sub a=2 into the equation:
b=-8

Now I'm having trouble finding c...
Any help would be greatly appreciated :)

Answer 1

If it has vertical asymptotes at x=1,4 then that means that
x^2+cx+4 = (x-1)(x-4)
So, c = -5