To find the equation of the oblique asymptote of the function [tex]\( g(x) = \frac{x^2 + x + 4}{x - 1} \)[/tex], we need to perform synthetic division to divide the polynomial [tex]\( x^2 + x + 4 \)[/tex] by [tex]\( x - 1 \)[/tex]. Below, you can find the detailed, step-by-step solution using synthetic division:
1. Set up the synthetic division:
- Write down the coefficients of the numerator polynomial [tex]\( x^2 + x + 4 \)[/tex], which are 1, 1, and 4.
- The denominator [tex]\( x - 1 \)[/tex] implies that we use the root [tex]\( x = 1 \)[/tex] in our synthetic division.
2. Write the initial setup:
[tex]\[
\begin{array}{r|rrr}
1 & 1 & 1 & 4 \\
\end{array}
\][/tex]
3. Perform the synthetic division:
- Bring down the leading coefficient (1 in this case).
[tex]\[
\begin{array}{r|rrr}
1 & 1 & 1 & 4 \\
& & 1 & \\
\hline
& 1 & & \\
\end{array}
\][/tex]
- Multiply this leading coefficient by the root (1) and write the result below the next coefficient.
[tex]\[
\begin{array}{r|rrr}
1 & 1 & 1 & 4 \\
& & 1 & 2 \\
\hline
& 1 & 2 & \\
\end{array}
\][/tex]
- Add the second original coefficient (1) to the product (1) to get 2 and write it below the line.
[tex]\[
\begin{array}{r|rrr}
1 & 1 & 1 & 4 \\
& & 1 & 2 \\
\hline
& 1 & 2 & \\
\end{array}
\][/tex]
- Repeat the process: multiply the last number below the line (2) by the root (1) and write the result below the next coefficient.
[tex]\[
\begin{array}{r|rrr}
1 & 1 & 1 & 4 \\
& & 1 & 2 \\
\hline
& 1 & 2 & 6 \\
\end{array}
\][/tex]
- Add the third original coefficient (4) to the product (2) to get 6.
The final row, except for the last number, gives us the coefficients of the quotient polynomial. Therefore, the quotient polynomial is [tex]\( x + 2 \)[/tex] and there is a remainder of 6.
4. Interpret the result:
- The quotient [tex]\( x + 2 \)[/tex] represents the polynomial part of the division, which approaches the value of [tex]\( g(x) \)[/tex] as [tex]\( x \)[/tex] becomes very large.
- Thus, the equation of the oblique asymptote is given by the quotient, [tex]\( y = x + 2 \)[/tex].
So, the equation of the oblique asymptote of [tex]\( g(x) = \frac{x^2 + x + 4}{x - 1} \)[/tex] is:
[tex]\[
\boxed{y = x + 2}
\][/tex]
