Problem 120 - Algebra and Trigonometry

Question

Find the slant asymptote of $$f(x)=\frac{2 x^{2}-7 x-1}{x-2}$$

Step-by-Step Solution

Answer

The slant asymptote of the function \(f(x)=\dfrac{2x^{2}-7x-1}{x-2}\) is \(2x-3\).

Step 1: Polynomial long division

Divide \(2x^{2}-7x-1\) by \(x-2\) using polynomial long division. The quotient is the desired slant asymptote. Ignore any remainder from the division.

Step 2: Perform the long division

Dividing, we begin by asking how many times does \(x\) go into \(2x^2\), which is \(2x\). Multiply \(2x\) by \(x-2\) to get \(2x^2-4x\). Subtract this from the original numerator to yield \(-3x-1\). Repeat this, asking how many times does \(x\) go into \(-3x\), which is \(-3\). Multiply \(-3\) by \(x-2\) to get \(-3x+6\). Subtract this from \(-3x-1\) to get \(-7\). Thus, the quotient of the long division is \(2x-3\).

Step 3: State slant asymptote

Thus, the slant asymptote of the function \(f(x)=\dfrac{2x^{2}-7x-1}{x-2}\) is \(2x-3\).

Key Concepts

Polynomial Long Division

Polynomial long division is a technique used to divide polynomials similar to how we divide numbers. We aim to find the quotient, ignoring any remainder, just as we do in numerical long division. For dividing the polynomial \(2x^{2}-7x-1\) by \(x-2\), start by figuring out how many times the divisor \(x\) fits into the first term \(2x^2\) of the dividend. The answer is \(2x\), which becomes the first term of our quotient. Multiply \(2x\) by the divisor \(x-2\) to get \(2x^2 - 4x\), and subtract this product from the dividend to find the remainder \(-3x - 1\).

Next, see how many times \(x\) fits into \(-3x\) – which is \(-3\). Repeat the multiplication and subtraction process with \(-3\), yielding a remainder of \(-7\). These repetitive steps allow us to systematically solve polynomials more complex than simple algebraic equations.

Rational Functions

Rational functions are fractions wherein both the numerator and the denominator are polynomials. In the function \(f(x)=\frac{2x^{2}-7x-1}{x-2}\), the numerator is \(2x^2-7x-1\) and the denominator is \(x-2\). These types of functions can have various characteristics, notably asymptotes, which indicate values that can make the function unbounded.

Understanding rational functions involves knowing how these asymptotes influence their graph, behavior, and limits. The division process we performed reveals more information about this rational expression, helping us interpret its graph by determining the slant asymptote.

Quotient

In polynomial division, the quotient is the result obtained when the divisor completely divides the dividend. The slant, or oblique, asymptote is typically represented by this resulting quotient. Looking at our exercise, the division of \(2x^{2}-7x-1\) by \(x-2\) results in a quotient of \(2x-3\).
This quotient is particularly significant as it provides the equation of the slant asymptote for the function. Therefore, the line \(y = 2x - 3\) acts as an approximation of the function for extreme values of \(x\). This helps understand how the function behaves beyond its standard domain.

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches or intersects. They are essential for describing the vertical and horizontal behavior of a rational function. In our function \(f(x)\), we are interested in finding a slant asymptote, which is a kind of asymptote that occurs when the degree of the numerator is higher than that of the denominator in a rational function.

Through polynomial long division, we've found this slant asymptote to be \(y = 2x - 3\). This means as \(x\) approaches positive or negative infinity, the graph of the rational function begins to resemble a straight line defined by this equation. Recognizing and understanding asymptotes allows us to graph rational functions accurately and anticipate their behavior at extremes of the domain.