To find the equation of the oblique asymptote of [tex]\( g(x) = \frac{x^2 + x + 4}{x - 1} \)[/tex], we can use synthetic division. Here's a detailed step-by-step solution:
1. Set up the synthetic division:
We have the numerator polynomial [tex]\( x^2 + x + 4 \)[/tex] and the denominator [tex]\( x - 1 \)[/tex]. Our synthetic division setup is: [tex]\( 1 \rfloor \quad 1 \quad 1 \quad 4 \)[/tex].
2. Perform synthetic division:
- Write down the coefficients of the numerator: [tex]\( 1 \)[/tex] (for [tex]\( x^2 \)[/tex]), [tex]\( 1 \)[/tex] (for [tex]\( x \)[/tex]), and [tex]\( 4 \)[/tex] (constant term).
- Start the division process. Bring down the first coefficient [tex]\( 1 \)[/tex].
[tex]\[
\begin{array}{r|rrr}
1 & 1 & 1 & 4 \\
& & 1 & 2 \\
\hline
& 1 & 1 & 6 \\
\end{array}
\][/tex]
Here's the step-by-step breakdown:
- Bring down the 1.
- Multiply 1 by 1 (divisor), result is 1. Write this under the next coefficient (i.e., under the 1).
- Add this 1 to the next coefficient (1), resulting in 2.
- Multiply this 2 by 1 (divisor), result is 2. Write this under the next coefficient (i.e., under the 4).
- Add this 2 to the next coefficient (4), resulting in 6.
3. Interpret the synthetic division result:
The quotient obtained from the synthetic division (disregarding the remainder) gives us the coefficients for the linear polynomial which represents the oblique asymptote.
The results from the synthetic division are: [tex]\(1, 2\)[/tex].
4. Form the equation of the oblique asymptote:
The oblique asymptote has the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are coefficients from the synthetic division.
From our synthetic division, we have:
- [tex]\( m = -1 \)[/tex]
- [tex]\( b = 0 \)[/tex]
5. Write the final equation:
Therefore, the equation of the oblique asymptote is:
[tex]\[
y = -x + 0 \quad \text{or simply} \quad y = -x.
\][/tex]
Hence, the equation of the oblique asymptote for [tex]\( g(x) = \frac{x^2 + x + 4}{x - 1} \)[/tex] is [tex]\( y = -x \)[/tex].
