Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 95

Chapter 4, Problem 95

Use long division to rewrite the equation for g in the form quotient + remainder/divisor. Then use this form of the function's equation and transformations of f(x) = 1/x to graph g. g(x) = (2x+7)/(x+3)

Verified step by step guidance

Identify the dividend and divisor for the long division. Here, the dividend is the numerator polynomial $2x + 7$ and the divisor is the denominator polynomial $x + 3$.

Set up the long division by dividing the leading term of the dividend (\$2x$) by the leading term of the divisor ($x$). This gives the first term of the quotient: $\frac{2x}{x} = 2$.

Multiply the entire divisor $x + 3$ by the quotient term $2$, resulting in $2(x + 3) = 2x + 6$.

Subtract this product from the original dividend: $(2x + 7) - (2x + 6)$. Simplify the subtraction to find the remainder.

Express the original function $g(x)$ as the quotient plus the remainder over the divisor: $g(x) = 2 + \frac{\text{remainder}}{x + 3}$. This form helps to analyze the graph by relating it to the parent function $f(x) = \frac{1}{x}$ with transformations.

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Key Concepts

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

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. It helps rewrite a rational function as a quotient plus a remainder over the divisor, simplifying the expression for analysis or graphing.

Rational Functions and Their Graphs

A rational function is a ratio of two polynomials. Understanding its graph involves identifying asymptotes, intercepts, and behavior at infinity, which can be made clearer by rewriting the function using long division.

Transformations of the Parent Function f(x) = 1/x

The function f(x) = 1/x is a basic rational function with a hyperbola shape. Graphing related functions involves applying transformations like shifts, stretches, and reflections to this parent graph based on the rewritten form of the function.

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In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-1) The graph passes through the point (-2,-3).

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In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. (1 − 3/(x+2)) / (1 + 1/(x−2))

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Solve each inequality in Exercises 86–91 using a graphing utility. 1/(x + 1) ≤ 2/(x + 4)

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