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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 85
Chapter 4, Problem 85

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2+x−6)/(x−3)

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Identify the given rational function: f(x)=x2+x−6x−3. We want to find the slant (oblique) asymptote, which occurs when the degree of the numerator is exactly one more than the degree of the denominator.
Perform polynomial long division of the numerator x^2 + x - 6 by the denominator x - 3. This will help us express f(x) as a quotient plus a remainder over the divisor, which reveals the slant asymptote.
Set up the division: divide the leading term of the numerator x^2 by the leading term of the denominator x to get the first term of the quotient. Then multiply the entire divisor by this term and subtract from the numerator. Repeat this process until the degree of the remainder is less than the degree of the divisor.
The quotient obtained from the division (ignoring the remainder) represents the equation of the slant asymptote. Write this quotient as a linear function y = mx + b where m and b come from the division result.
For graphing, use the seven-step strategy: (1) find the domain, (2) find intercepts, (3) find vertical asymptotes (where denominator is zero), (4) find the slant asymptote from the division, (5) analyze end behavior using the slant asymptote, (6) plot points around the vertical asymptote and near the slant asymptote, and (7) sketch the graph showing the behavior near asymptotes and intercepts.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the behavior of rational functions, including their domains and discontinuities, is essential for analyzing their graphs and asymptotes.
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Intro to Rational Functions

Slant (Oblique) Asymptotes

Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They represent the line that the graph approaches as x approaches infinity or negative infinity, found by performing polynomial division.
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Introduction to Asymptotes

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. It helps find the quotient and remainder, where the quotient gives the equation of the slant asymptote for rational functions with numerator degree one higher than denominator.
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Introduction to Polynomials
Related Practice
Textbook Question

Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x4−3x2−4). If f(3)=−150, determine the value of a_n.

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Textbook Question

Solve each inequality in Exercises 86–91 using a graphing utility. 2x2 + 5x - 3 ≤ 0

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Textbook Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x3+1)/(x2+2x)

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Textbook Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2+1)/x

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Textbook Question

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x+6=0

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Textbook Question

Solve each inequality in Exercises 86–91 using a graphing utility. x2 + 3x - 10 > 0

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