Oblique Asymptote: Find K For F(x) = (9x^2 + 36x + 41) / (3x + 5)

Hey guys! Today, we're diving into a super interesting problem involving oblique asymptotes. We're given a function, and we need to figure out a specific value related to its asymptote. Let's break it down step by step so it's crystal clear. Our mission, should we choose to accept it, is to find the value of 'k' in the oblique asymptote equation y = 3x + k for the function f(x) = (9x^2 + 36x + 41) / (3x + 5). Sounds like fun, right? Let's get started!

Understanding Oblique Asymptotes

First things first, let's make sure we're all on the same page about what an oblique asymptote actually is. Think of it as a line that a curve approaches as x heads towards infinity (either positive or negative). It's like the curve's guiding rail way out on the edges of the graph. Now, when do these asymptotes show up? Oblique asymptotes appear in rational functions (that's fractions where both the numerator and denominator are polynomials), specifically when the degree of the numerator is exactly one more than the degree of the denominator. In our case, f(x) = (9x^2 + 36x + 41) / (3x + 5), the numerator (9x^2 + 36x + 41) has a degree of 2, and the denominator (3x + 5) has a degree of 1. Bingo! That's our signal that we've got an oblique asymptote situation on our hands.

Why is this important? Well, knowing we're dealing with an oblique asymptote lets us use a specific method to find its equation. The general form of an oblique asymptote is y = mx + b, where m is the slope and b is the y-intercept. In our problem, we're told that the oblique asymptote is y = 3x + k, so we already know the slope (m = 3), and we just need to find k. The secret weapon for finding oblique asymptotes is polynomial long division. Yes, it might sound a bit scary, but trust me, it's just a systematic way to rewrite our rational function into a more helpful form.

Finding the Oblique Asymptote Using Polynomial Long Division

Okay, let's roll up our sleeves and tackle the polynomial long division. We're going to divide (9x^2 + 36x + 41) by (3x + 5). If you haven't done this in a while, no worries, we'll go through it step by step. It's really similar to regular long division with numbers, just with polynomials instead. First, we set up the division like this:

             ____________________
3x + 5   |   9x^2 + 36x + 41

Now, we ask ourselves: what do we need to multiply 3x by to get 9x^2? The answer is 3x. So, we write 3x above the line, lined up with the x term:

             3x_________________
3x + 5   |   9x^2 + 36x + 41

Next, we multiply (3x + 5) by 3x, which gives us 9x^2 + 15x. We write this below the dividend (9x^2 + 36x + 41) and subtract:

             3x_________________
3x + 5   |   9x^2 + 36x + 41
            -(9x^2 + 15x)
            ____________________
                    21x + 41

Now, we bring down the +41. We repeat the process: what do we need to multiply 3x by to get 21x? The answer is 7. So, we add +7 to our quotient above the line:

             3x + 7____________
3x + 5   |   9x^2 + 36x + 41
            -(9x^2 + 15x)
            ____________________
                    21x + 41

Multiply (3x + 5) by 7, which gives us 21x + 35. Subtract this from 21x + 41:

             3x + 7____________
3x + 5   |   9x^2 + 36x + 41
            -(9x^2 + 15x)
            ____________________
                    21x + 41
            -(21x + 35)
            ____________________
                            6

We're left with a remainder of 6. So, we can rewrite our original function f(x) as:

f(x) = 3x + 7 + 6 / (3x + 5)

This is the key step! Notice that we've separated the function into a polynomial part (3x + 7) and a remainder part (6 / (3x + 5)).

Identifying the Oblique Asymptote Equation

Now, let's zoom in on what happens as x gets super large (either positive or negative). The term 6 / (3x + 5) becomes incredibly small, approaching zero. Why? Because we're dividing a constant (6) by a very large number (3x + 5). Think of it like splitting a pizza among billions of people – each person gets almost nothing! So, as x goes to infinity, the function f(x) gets closer and closer to just 3x + 7. And there you have it – the oblique asymptote is y = 3x + 7. The polynomial part of our result from the long division is the equation of the oblique asymptote. This is the crucial takeaway from the polynomial long division process.

We were given that the oblique asymptote is y = 3x + k, and we've just found that it's y = 3x + 7. Can you see where this is going? By comparing the two equations, it's clear that k must be equal to 7. High five! We've solved the problem!

Determining the Value of k

So, to recap, we've done the heavy lifting. We performed polynomial long division on f(x) = (9x^2 + 36x + 41) / (3x + 5) and rewrote it as f(x) = 3x + 7 + 6 / (3x + 5). We reasoned that as x approaches infinity, the term 6 / (3x + 5) goes to zero, leaving us with the oblique asymptote y = 3x + 7. Now, we simply match this up with the given form of the asymptote, y = 3x + k. By comparing the two, we can confidently say that k = 7. It's like finding the missing piece of a puzzle! We've successfully identified the value of k by understanding the behavior of the function as x becomes very large and using polynomial long division to isolate the oblique asymptote.

Conclusion

Alright, guys, we did it! We successfully navigated the world of oblique asymptotes and found that the value of k is 7. This problem might have seemed a bit daunting at first, but by breaking it down into smaller steps – understanding oblique asymptotes, using polynomial long division, and carefully comparing equations – we were able to solve it. Remember, math is all about taking complex problems and making them manageable. I hope this explanation was helpful and that you feel more confident tackling similar problems in the future. Keep practicing, keep exploring, and most importantly, keep having fun with math!