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MAT 151: College Algebra with Trigonometry

Section 1.3 Operations on Rational Expressions

Goals:
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  • Be able to simplify and perform operations on rational expressions.
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Problem 1.3.1.

Let \(f(x)=\frac{2x^2-3x-2}{x-2}\text{.}\)
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  1. In Desmos, graph \(f(x)\text{.}\) Is this graph what you expected? Sketch the graph of \(f(x)\) below. Write an equation describing \(f(x)\) based on this graph.
    A coordinate grid showing the x- and y-axes with arrowheads, labeled in increments of 5 from -10 to 10. The origin \((0, 0)\) is marked at the center. Light square grid lines fill the plane; the axes are bold.
    Figure 1.3.2. Blank coordinate plane with grid and labeled axes.
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  2. Rewrite \(f(x)\) by factoring its numerator and simplifying.
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  3. How are \(f(x)=\frac{2x^2-3x-2}{x-2}\) and the expression you found in Item 2 the same? How are they different?
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Problem 1.3.3.

Simplify each rational expression.
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  1. \(\displaystyle \frac{3xy-15y}{6xy+12y}\)
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  2. \(\displaystyle \frac{x^2+8x+12}{x^2+3x-18}\)
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Problem 1.3.4.

Decide which operation each problem below uses (addition, subtraction, multiplication or division of fractions), then solve the problem.
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  1. Stacey eats \(\frac{1}{3}\) of a candy bar she just bought. Her friend asks her for some, so she gives her friend \(\frac{3}{4}\) of her remaining candy bar. How much of the original candy bar is this?
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  2. A baker has a cake recipe that calls for \(\frac{1}{3}\) of a cup of sugar and a brownie recipe that calls for \(\frac{3}{4}\) of a cup of sugar. How much sugar does he need for both recipes?
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  3. You and two friends buy a pretzel. One of your friends takes \(\frac{1}{3}\) of the whole pretzel, and the other takes \(\frac{1}{2}\) of the whole pretzel. How much of the whole pretzel did they leave you?
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  4. You and some friends chip-in to buy \(\frac{1}{2}\) of a pie from the school bake sale. You decide that each person who chips-in should get a slice that is \(\frac{1}{8}\) of a whole pie. How many people should contribute to buying the pie?
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Problem 1.3.5.

Perform the indicated operation and write your answer in simplest form.
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  1. \(\displaystyle \frac{r^3s}{t} \div \frac{rs^3}{t^3}\)
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  2. \(\displaystyle \frac{5x^2}{x+2} \div \frac{10}{x^2+2x}\)
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  3. \(\displaystyle \frac{y^2+10y+25}{y^2-9} \cdot \frac{y^2+3y}{y+5}\)
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Problem 1.3.6.

Perform the indicated operation and write your answer in simplest form.
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  1. \(\displaystyle \frac{6}{x}+\frac{7}{5x^2}\)
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  2. \(\displaystyle \frac{x-1}{x-2}+\frac{x^2+4x-4}{x^2+4x-12}\)
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  3. \(\displaystyle \frac{2x-5}{x-3} - \frac{2x-1}{x-1}\)
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Problem 1.3.7.

Let \(f(x)=\frac{x-2}{x^2-x-6}\) and \(g(x)=\frac{3x^2-12}{(x-1)(x-3)}\text{.}\)
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  1. Express \(f(x)+g(x)\) as a single fraction.
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  2. Express \(f(x)-g(x)\) as a single fraction.
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  3. Express \(\frac{f(x)}{g(x)}\) as a single fraction.
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NOTE: Whenever a rational expression has a numerator with degree equal or greater than the degree of the denominator, long division of polynomials can be used to produce an equivalent expression with a polynomial and a rational expression, where the numerator in the rational expression has smaller degree than the denominator.
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Problem 1.3.8.

Use long division to rewrite \(\frac{x}{x-3}\text{.}\)
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Problem 1.3.9.

Use long division to rewrite \(\frac{x^2+7x+8}{x-2}\text{.}\)
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Problem 1.3.10.

There are numbers \(a\) and \(b\) such that \(\frac{1}{x^2-x-6}=\frac{a}{x-3}+\frac{b}{x+2}\text{.}\) Find the values of \(a\) and \(b\text{.}\)
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