Find the Asymptotes f(x)=(x^3-8x^2+15x)/(x^2-5x)

Solution:

Step 1: Identify Undefined Points

Determine the values for which the function $\frac{x^3-8x^2+15x}{x^2-5x}$ is not defined. These are $x=0$ and $x=5$.

Step 2: Vertical Asymptotes

Check for infinite discontinuities to find vertical asymptotes. There are none in this case.

Step 3: Horizontal Asymptotes Rules

For a rational function $R(x)=\frac{a{x}^n}{b{x}^m}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator:

• If $n<m$, the horizontal asymptote is $y=0$.

• If $n=m$, the horizontal asymptote is $y=\frac{a}{b}$.

• If $n>m$, there is no horizontal asymptote, but potentially an oblique asymptote.

Step 4: Determine Degrees

Calculate the degrees of the numerator and denominator: $n=3$ and $m=2$.

Step 5: Horizontal Asymptote Conclusion

Since $n>m$, there is no horizontal asymptote.

Step 6: Oblique Asymptote Calculation

Perform polynomial long division to find the oblique asymptote.

Step 6.1: Simplify the Function

Begin simplifying the function.

Step 6.1.1: Simplify the Numerator

Start with the numerator $x^3-8x^2+15x$.

Step 6.1.1.1: Factor Out $x$

Extract $x$ from each term to get $x(x^2-8x+15)$.

Step 6.1.1.2: Factor the Quadratic

Use the AC method to factor $x^2-8x+15$.

Step 6.1.1.2.1: Find Integers for AC Method

Identify integers with a product of $15$ and a sum of $-8$, which are $-5$ and $-3$.

Step 6.1.1.2.2: Write Factored Form

Express the factored form as $x((x-5)(x-3))$.

Step 6.1.2: Simplify the Denominator

Now, focus on the denominator $x^2-5x$.

Step 6.1.2.1: Factor Out $x$

Factor $x$ from the denominator to get $x(x-5)$.

Step 6.1.2.2: Cancel Common Factors

Cancel the common $x$ factor.

Step 6.1.2.2.1: Perform Cancellation

After canceling, the function simplifies to $\frac{(x-5)(x-3)}{x-5}$.

Step 6.1.2.3: Cancel Additional Common Factors

Cancel the common $(x-5)$ factor.

Step 6.1.2.3.1: Final Simplification

The final simplified form is $x-3$.

Step 6.2: Determine the Oblique Asymptote

The oblique asymptote is given by the simplified polynomial, which is $y=x-3$.

Step 7: Asymptotes Summary

Summarize all asymptotes found:

• No Vertical Asymptotes
• No Horizontal Asymptotes
• Oblique Asymptote: $y=x-3$

Step 8: End of Solution Process

Knowledge Notes:

The process involves several key concepts in algebra and calculus:

1. Undefined Points: Points where the function is not defined due to division by zero.

2. Vertical Asymptotes: Lines where the function approaches infinity. They occur at the undefined points unless there is a common factor in the numerator and denominator that cancels out.

3. Horizontal Asymptotes: Lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. They depend on the degrees of the polynomial in the numerator ($n$) and denominator ($m$).

4. Oblique (Slant) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. They can be found by performing polynomial long division.

5. Polynomial Long Division: A method used to divide polynomials, similar to long division with numbers.

6. Factoring: The process of breaking down a composite number or polynomial into a product of other numbers or polynomials that, when multiplied, give the original number or polynomial.

7. AC Method: A factoring technique used to factor trinomials.

8. Cancellation: The process of reducing a fraction by eliminating common factors from the numerator and the denominator.