SOLUTION: I need to find the slant asymptotes of a. x^3+x^2/x^2-4 b. 2x^2+10x-12/x^2+x-6 I understand how to find the Vertical and Horizontal, but I'm confused on the slant portion.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I need to find the slant asymptotes of a. x^3+x^2/x^2-4 b. 2x^2+10x-12/x^2+x-6 I understand how to find the Vertical and Horizontal, but I'm confused on the slant portion.       Log On


   

Question 734948: I need to find the slant asymptotes of
a. x^3+x^2/x^2-4
b. 2x^2+10x-12/x^2+x-6
I understand how to find the Vertical and Horizontal, but I'm confused on the slant portion. Please help! thank you!!!
Found 2 solutions by lynnlo, MathLover1:
Answer by lynnlo(4176) About Me  (Show Source):
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
to find the slant asymptotes of
a. %28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29
x%5E2%28x%2B1%29%2F%28x%5E2-2%5E2%29
x%5E2%28x%2B1%29%2F%28%28x-2%29%28x%2B2%29%29
The vertical asymptotes (and any restrictions on the domain) come from the zeroes of the denominator, so I'll set the denominator equal to zero and solve.
%28%28x-2%29%28x%2B2%29%29=0
if x-2=0...=>...x=2
if x%2B2=0...=>...x=-2
Then the domain is all x-values other than ± 2, and the two vertical asymptotes are at
x+=+2 and x=-2.
Since the degree is greater in the denominator than in the numerator, the y-values will be dragged down to the x-axis, and the horizontal asymptote is therefore "y+=+0" (the x-axis) .
The slant asymptote: use the long division to find it
----------x%2B1
--------|x%5E3%2Bx%5E2
x%5E2-4)..x%5E3-4x
_________________
----------0%2B4x%2Bx%5E2
------------ x%5E2-4
________________
---------------4x%2B4
---------------4%28x%2B1%29
so, since x%2B1 is a factor of the numerator
the slant asymptote: y+=+x+%2B+1



b. %282x%5E2%2B10x-12%29%2F%28x%5E2%2Bx-6%29
horizontal asymptote: -> to -12%2F-6=2 as x-> infinity or -infinity
so,horizontal asymptote is y=2
continue factoring
%282x%5E2%2B10x-12%29%2F%28x%5E2%2Bx-6%29+
2%28x%5E2%2B5x-6%29%2F%28x%5E2%2Bx-6%29+
2%28x%5E2-x%2B6x-6%29%2F%28x%5E2%2B3x-2x-6%29+
2%28%28x%5E2-x%29%2B%286x-6%29%29%2F%28%28x%5E2%2B3x%29-%282x%2B6%29%29+
2%28x%28x-1%29%2B6%28x-1%29%29%2F%28x%28x%2B3%29-2%28x%2B3%29%29+
2%28%28x%2B6%29%28x-1%29%29%2F%28%28x-2%29%28x%2B3%29%29+
find values that make denominator equal to zero:
ifx-2=0+....->...x=2
ifx%2B3=0+....->...x=-3
these are vertical asymptotes
-there is no slant asymptote in this case