# What is the oblique asymptote of y = ( x^3 + 5x^2 + 3x + 10 )/( x^2 + 1 )?

Sep 2, 2015

The oblique asymptote is $\textcolor{red}{y = x + 5}$

#### Explanation:

$y = \frac{{x}^{3} + 5 {x}^{2} + 3 x + 10}{{x}^{2} + 1}$

A slant (oblique) asymptote occurs when the polynomial in the numerator is a higher degree than the polynomial in the denominator.

To find the slant asymptote you must divide the numerator by the denominator.

I will use synthetic division:

$\text{ "" "|1" "5" "3" } 10$
$\textcolor{w h i t e}{1} - 1 | \text{ } \textcolor{w h i t e}{1} - 1 \textcolor{w h i t e}{1} - 5$
$0 \text{ "color(white)(1)|" "0" } 0$
" "" "stackrel("—————————)
" "" "color(white)(1)1" "5" "color(red)(2" "color(white)(1)5)

The quotient is $x + 5$ with a remainder of $2 x + 5$.

We can ignore the remainder, so the oblique asymptote is $y = x + 5$.