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Vertical asymptotes occur at the values of x that make the denominator of a rational function equal to zero. These are important because they indicate points where the function goes to infinity. To identify a vertical asymptote, set the denominator to zero and solve for x. For the function \(k(x)=\frac{-2 x^{2}-3 x+7}{x+3}\), set \(x + 3 = 0\). Solving for x, we get \(x = -3\). This means there is a vertical asymptote at \(x = -3\). This asymptote tells us that as the function approaches -3, the values of k(x) increase or decrease without bound.

Horizontal asymptotes describe the behavior of a function as \(x\) tends to infinity or negative infinity. They are found by comparing the degrees of the numerator and the denominator. For a rational function, the rules are as follows:
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is \( y = \frac{a}{b} \) where \(a\) and \(b\) are the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
In the given function \(k(x)=\frac{-2 x^{2}-3 x+7}{x+3}\), the degree of the numerator (2) is more than the degree of the denominator (1). Because of this, there is no horizontal asymptote.

Oblique or slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They provide a linear approximation to the function as x goes to infinity or negative infinity. To find an oblique asymptote, you use polynomial long division to divide the numerator by the denominator.
For our function, the numerator is \(-2x^2 - 3x + 7\) and the denominator is \(x + 3\). Performing polynomial long division, we get:
- Divide \(-2x^2 \) by \(x\): \(-2x\).
- Multiply \(-2x\) by \((x + 3)\): \(-2x^2 - 6x\).
- Subtract this from the numerator: \(( -2x^2 - 3x + 7) - ( -2x^2 - 6x ) = 3x + 7\).
- Divide \(3x\) by \(x\): \(3\).
- Multiply \(3\) by \(x + 3\): \(3x + 9\).
- Subtract from the remaining polynomial: \(( 3x + 7) - ( 3x + 9 ) = -2\).
So our quotient is \(-2x + 3\). Therefore, the oblique asymptote is \(y = -2x + 3\).

Polynomial long division is a method for simplifying a complex polynomial by dividing it by another polynomial of a lower degree, similar to long division with numbers. This technique is essential for identifying oblique asymptotes in rational functions.
Here’s how it works:

• Divide the first term of the numerator by the first term of the denominator.
• Multiply the entire denominator by this result and subtract from the numerator.
• Repeat this process with the new polynomial formed after subtraction until the degree of the remaining polynomial is less than the degree of the denominator.

For example, in dividing \(-2x^2 - 3x + 7\) by \(x + 3\), you would begin by dividing \(-2x^2\) by \(x\) to get \(-2x\), continue this process, and finally obtain the quotient \(-2x + 3\), which can then be used to identify the oblique asymptote \(y = -2x + 3\).