SOLUTION: find the vertical, horizontal, and slant asymptotes, if any, for f(x)=5x^3+49x^2+127x+75/x^2+9x+18

Algebra ->  Rational-functions -> SOLUTION: find the vertical, horizontal, and slant asymptotes, if any, for f(x)=5x^3+49x^2+127x+75/x^2+9x+18      Log On


   

Question 790305: find the vertical, horizontal, and slant asymptotes, if any, for f(x)=5x^3+49x^2+127x+75/x^2+9x+18
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
You meant
f(x)=(5x^3+49x^2+127x+75)/(x^2+9x+18)=%285x%5E3%2B49x%5E2%2B127x%2B75%29%2F%28x%5E2%2B9x%2B18%29

Factoring, we get x%5E2%2B9x%2B18=%28x%2B6%29%28x%2B3%29,
and 5x%5E3%2B49x%5E2%2B127x%2B75 is divisible by x%2B3
5x%5E3%2B49x%5E2%2B127x%2B75=%285x%5E2%2B34x%2B25%29%28x%2B3%29, so

So, highlight%28x=-3%29 and highlight%28x=-6%29 are vertical asymptotes,
and for x%3C%3E-3, f%28x%29=%285x%5E2%2B34x%2B25%29%2F%28x%2B6%29

If we divide 5x%5E2%2B34x%2B25 by x%2B6,
the quotient is 5x%2B4 and the remainder is 1.
In other words 5x%5E2%2B34x%2B25=%28x%2B6%29%285x%2B4%29%2B1
So for x%3C-6 and for x%3E0,
approaches y=5x%2B4 as abs%28x%29 increases.
So highlight%28y=5x%2B4%29 is the slant asymptote on both sides.

IMPORTANT NOTE:
Those parentheses are essential if you cannot type the equation as the "two-story" version,
f%28x%29=%285x%5E3%2B49x%5E2%2B127x%2B75%29%2F%28x%5E2%2B9x%2B18%29, with a long horizontal line in the middle.
The horizontal line implies the parentheses.
Without the parentheses, you get
f(x)=5x^3+49x^2+127x+75/x^2+9x+18=5x%5E3%2B49x%5E2%2B127x%2B75%2Fx%5E2%2B9x%2B18