To find the value of [tex]\( k \)[/tex] for the oblique asymptote of the rational function [tex]\( f(x) = \frac{9x^2 + 36x + 41}{3x + 5} \)[/tex], we need to perform polynomial long division.
### Step-by-Step Solution:
Step 1: Identify the degrees of the numerator and denominator.
- The degree of the numerator [tex]\( 9x^2 + 36x + 41 \)[/tex] is 2.
- The degree of the denominator [tex]\( 3x + 5 \)[/tex] is 1.
Since the degree of the numerator is one more than the degree of the denominator, there is an oblique asymptote.
Step 2: Perform polynomial long division:
1. First division:
- Divide the leading term of the numerator by the leading term of the denominator: [tex]\(\frac{9x^2}{3x} = 3x\)[/tex].
- Multiply the entire denominator by this result: [tex]\(3x \cdot (3x + 5) = 9x^2 + 15x\)[/tex].
- Subtract this product from the original numerator:
[tex]\[
(9x^2 + 36x + 41) - (9x^2 + 15x) = 21x + 41.
\][/tex]
2. Second division:
- Again, divide the leading term of the current result by the leading term of the denominator: [tex]\(\frac{21x}{3x} = 7\)[/tex].
- Multiply the whole denominator by 7: [tex]\(7 \cdot (3x + 5) = 21x + 35\)[/tex].
- Subtract this product from the result of the previous step:
[tex]\[
(21x + 41) - (21x + 35) = 6.
\][/tex]
Step 3: Determine the quotient and remainder:
- Quotient: [tex]\( 3x + 7 \)[/tex]
- Remainder: [tex]\( 6 \)[/tex]
The oblique asymptote is given by the quotient of the division, which is [tex]\( y = 3x + 7 \)[/tex].
Thus, the value of [tex]\( k \)[/tex] is [tex]\( \boxed{7} \)[/tex].
