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Chapter 3: Problem 50

Identify the asymptotes. $$ q(x)=\frac{x^{3}+3 x^{2}-2 x-4}{x^{2}-7} $$

Short Answer

Expert verified
Vertical asymptotes: \[x = \pm\sqrt{7}\]. No horizontal asymptotes. There is an oblique asymptote.

Step by step solution

01

- Identify Vertical Asymptotes

To find the vertical asymptotes, set the denominator equal to zero and solve for x. Solve: \[x^2 - 7 = 0\] This gives: \[x = \pm\sqrt{7}\]
02

- Identify Horizontal Asymptotes

Compare the degrees of the numerator and the denominator. Since the degree of the numerator (3) is greater than the degree of the denominator (2), the horizontal asymptote does not exist.
03

- Identify Oblique Asymptotes

Since the degree of the numerator (3) is exactly 1 more than the degree of the denominator (2), there is an oblique asymptote. Perform polynomial long division to find the oblique asymptote: \[\frac{x^3+3x^2-2x-4}{x^2-7}\] The result of the division will be the equation of the oblique asymptote.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertical asymptotes
Vertical asymptotes occur where the function goes to infinity. To find them, set the denominator of the rational function equal to zero and solve for x.
In our exercise, we solve the equation
\[ x^2 - 7 = 0 \]
for x.
This will give us:
\[ x = \pm \sqrt{7} \]
The values \sqrt{7} and -\sqrt{7} are the points where vertical asymptotes occur.
Remember, the function cannot be evaluated at these points.
horizontal asymptotes
Horizontal asymptotes provide information about the behavior of the function at extreme values of x (as x approaches positive or negative infinity).
To determine if a rational function has a horizontal asymptote, compare the degrees of the numerator and the denominator.
  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
  • If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
  • If the degree of the numerator is greater than the degree of the denominator, no horizontal asymptote exists.
In this exercise, the degree of the numerator (3) is greater than the degree of the denominator (2), so there is no horizontal asymptote.
oblique asymptotes
Oblique asymptotes, also known as slant asymptotes, occur when the degree of the numerator is exactly 1 more than the degree of the denominator. This gives the function an asymptotic behavior that is linear as x goes to infinity.
To find the oblique asymptote, perform polynomial long division on the rational function.
In our example, we divide:
\[\frac{x^3 + 3x^2 - 2x - 4}{x^2 - 7}\]
The quotient from this division will be the equation of the oblique asymptote. This division might be complex, so practice polynomial long division to become proficient in finding oblique asymptotes.
polynomial long division
Polynomial long division is a method used to divide polynomials, similar to the long division we use for numbers.
Here's how you can perform polynomial long division:
  • Write the division in standard form.
  • Divide the first term of the numerator by the first term of the denominator.
  • Multiply the entire denominator by this result and subtract it from the numerator.
  • Repeat the process with the new polynomial obtained after subtraction.
Let鈥檚 apply this method to our example:
1. Divide \(x^3\) by \(x^2\) to get \(x\).
2. Multiply \(x\) by \(x^2 - 7\) to get \(x^3 - 7x\).
3. Subtract \(x^3 - 7x\) from \(x^3 + 3x^2 - 2x - 4\) to get \(3x^2 + 5x - 4\).
4. Now, divide \(3x^2\) by \(x^2\) to get \(3\).
5. Multiply \(3\) by \(x^2 - 7\) to get \(3x^2 - 21\).
6. Subtract this from \(3x^2 + 5x - 4\) to get \(5x + 17\).
This gives us:
\[ q(x) = x + 3 + \frac{5x + 17}{x^2 - 7} \]
The term \(x + 3\) is the oblique asymptote.

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