Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the oblique asymptote for $$ f(x)=\frac{3 x^{3}+4 x^{2}-x+1}{x^{2}+1} $$

Step-by-Step Solution

Answer

The oblique asymptote is \(y = 3x + 4\).

Step 1: Understand the conditions for an oblique asymptote

An oblique (or slant) asymptote occurs when the degree of the polynomial in the numerator is exactly one degree higher than the degree of the polynomial in the denominator. In this case, the numerator is a cubic polynomial with a degree of 3, and the denominator is a quadratic polynomial with a degree of 2.

Step 2: Perform polynomial long division

To find the oblique asymptote, divide the numerator \(3x^3 + 4x^2 - x + 1\) by the denominator \(x^2 + 1\). Perform polynomial long division until the degree of the remainder is less than the degree of the denominator.

Step 3: Polynomial long division - first term

Take the first term of the numerator \(3x^3\) and divide it by the first term of the denominator \(x^2\) to get \(3x\). This is the first term of the quotient.

Step 4: Subtract the result

Multiply \(3x\) by the entire denominator \(x^2 + 1\), which results in \(3x^3 + 3x\). Subtract this from the original numerator to find the new polynomial equation: \(4x^2 - 4x + 1\).

Step 5: Polynomial long division - second term

Now, take the first term of the new polynomial \(4x^2\) and divide by \(x^2\) to get \(4\). Add this to the quotient, resulting in \(3x + 4\).

Step 6: Subtract the result again

Multiply \(4\) by the denominator \(x^2 + 1\), giving \(4x^2 + 4\). Subtract this from \(4x^2 - 4x + 1\), leaving \(-4x - 3\).

Step 7: Identify the oblique asymptote

The quotient \(3x + 4\) is the oblique asymptote of the function, since the division's remainder degree \(-4x - 3\) is less than the degree of the denominator.

Key Concepts

Polynomial Long Division

Polynomial long division is a technique similar to numerical long division but applied to polynomials. It is used to divide one polynomial by another to find the quotient and remainder.
To perform polynomial long division:

• First, compare the degree of the term you are dividing to the leading term of the divisor (the polynomial on the bottom).
• Then, divide the leading term of the numerator by the leading term of the denominator.
• Write the result above the division line, and multiply the entire divisor by this result.
• Subtract the product from the original polynomial.
• Repeat this process with the resulting polynomial until the degree of the remainder is less than the degree of the divisor.

This method helps in cases like calculating oblique asymptotes where rational functions are considered.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in its expression. It is a crucial factor in determining the nature of polynomial equations.
For example, in the polynomial \(3x^3 + 4x^2 - x + 1\), the degree is 3.

• The degree tells us about the number of roots the polynomial could have and the shape of its graph.
• In the context of rational functions, knowing the degree of the numerator and the denominator is essential in identifying asymptotic behavior.

The degree plays a critical role in deciding whether the polynomial has aspects like horizontal or oblique asymptotes.

Cubic Polynomial

A cubic polynomial is a polynomial of degree three, meaning the highest exponent of its variable \(x\) is three.
Cubic polynomials generally have the form \(ax^3 + bx^2 + cx + d\).

• They can have up to three real roots.
• The graph of a cubic polynomial is characterized by its potential to have one or two turning points.
• Cubic polynomials can be used to model situations involving acceleration and other physical processes.

In rational functions, if the numerator is a cubic polynomial and the denominator’s degree is 2, as in our exercise, they can result in an oblique asymptote.

Rational Functions

Rational functions are ratios of two polynomials. They are written in the form \(\frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomials.
Key characteristics of rational functions include:

• Their domains exclude values that make the denominator zero.
• They can have vertical asymptotes where the denominator is zero, horizontal or oblique asymptotes depending on the degrees of the numerator and denominator.
• The simplification of rational functions often involves polynomial long division, especially if finding asymptotes is necessary.

Understanding rational functions is important for analyzing complex graphs and solving equations involving rates and proportions.