Rational Expressions and Problem Solving in College Algebra

Section 14.1: Rational Functions and Simplifying Rational Expressions

Find the Domain of a Rational Function

A rational function is any function that can be written as $f(x) = \frac{P(x)}{Q(x)}$, where P and Q are polynomials and $Q(x) \neq 0$. The domain of a rational function is all real numbers except those that make the denominator zero.

• Key Point: To find the domain, set the denominator equal to zero and solve for excluded values.

• Example: $f(x) = \frac{3x}{7-x}$; domain is all real numbers except $x = 7$.

Simplify or Write Rational Expressions in Lowest Terms

To simplify a rational expression, factor both numerator and denominator and cancel common factors.

• Key Point: Only factors (not terms) can be cancelled.

• Example: $\frac{x^2-9}{x^2-6x+9} = \frac{(x-3)(x+3)}{(x-3)^2} = \frac{x+3}{x-3}$, $x \neq 3$.

Fundamental Principle of Rational Expressions

For any rational expression $\frac{PR}{QR}$ and any polynomial $R$, $R \neq 0$:

Original Expression	Simplified Form
$\frac{PR}{QR}$	$\frac{P}{Q}$

Write Equivalent Rational Expressions

Equivalent rational expressions can be written as $\frac{-a}{b} = \frac{a}{-b} = -\frac{a}{b}$.

• Key Point: Multiplying numerator or denominator by -1 changes the sign of the fraction.

• Example: $\frac{x-11}{x-4}$ is equivalent to $-\frac{11-x}{x-4}$.

Use Rational Functions in Applications

Rational functions are used to model real-world scenarios, such as revenue, rates, and proportions.

• Example: If $R(t) = \frac{100t^2}{t^2+4}$ models revenue, find values by substituting $t$.

Section 14.2: Multiplying and Dividing Rational Expressions

Multiply Rational Expressions

To multiply rational expressions, multiply numerators and denominators:

• Formula: $\frac{P}{Q} \times \frac{R}{S} = \frac{PR}{QS}$

• Steps:

  1. Factor all numerators and denominators.

  2. Cancel common factors.

  3. Multiply remaining factors.

• Example: $\frac{3}{10} \times \frac{2x+10}{x+5}$

Divide Rational Expressions

To divide rational expressions, multiply by the reciprocal of the divisor:

• Formula: $\frac{P}{Q} \div \frac{R}{S} = \frac{P}{Q} \times \frac{S}{R} = \frac{PS}{QR}$, $R \neq 0$

• Example: $\frac{x+2}{7-x} \div \frac{x^2-5x+6}{7-x}$

Convert Between Units of Measure

Rational expressions can be used to convert units, such as cubic yards to cubic feet.

• Example: $3$cubic yards$= 3 \times 27 = 81$ cubic feet

Section 14.3: Adding and Subtracting Rational Expressions with Common Denominators and Least Common Denominator

Add and Subtract Rational Expressions with the Same Denominator

When denominators are the same, add or subtract numerators:

• Formula: $\frac{P}{Q} + \frac{R}{Q} = \frac{P+R}{Q}$

• Example: $\frac{2}{3} + \frac{5}{3} = \frac{7}{3}$

Add and Subtract Rational Expressions with Common Denominators

For expressions with common denominators, combine numerators and keep the denominator.

• Example: $\frac{2x+3}{3x+9} - \frac{x-2}{3x+9} = \frac{2x+3-(x-2)}{3x+9}$

Find the Least Common Denominator (LCD)

To add or subtract rational expressions with different denominators, find the LCD.

• Steps:

  1. Factor each denominator.

  2. Identify all unique factors.

  3. Multiply each factor the greatest number of times it occurs in any denominator.

• Example: LCD of $3m^2+15$ and $m+5$ is $(m+5)(3m-15)$

Write a Rational Expression as an Equivalent Expression Whose Denominator is Given

Rewrite a rational expression so that it has a specified denominator by multiplying numerator and denominator by necessary factors.

• Example: $\frac{6}{3a}$ as denominator $12ab^2$ becomes $\frac{6 \times 4b^2}{3a \times 4b^2} = \frac{24b^2}{12ab^2}$

Section 14.4: Adding and Subtracting Rational Expressions with Unlike Denominators

Add and Subtract Rational Expressions with Unlike Denominators

To add or subtract rational expressions with different denominators:

• Steps:

  1. Find the LCD of the denominators.

  2. Rewrite each expression with the LCD as the denominator.

  3. Add or subtract the numerators.

  4. Simplify the result if possible.

• Example: $\frac{5x+7}{x}$ and $\frac{4x}{x^2}$; LCD is $x^2$.

Section 14.5: Solving Equations Containing Rational Expressions

Solve Equations Containing Rational Expressions

To solve equations with rational expressions:

• Steps:

  1. Find the LCD of all denominators.

  2. Multiply both sides by the LCD to clear denominators.

  3. Solve the resulting equation.

  4. Check for extraneous solutions (values that make any denominator zero).

• Example: $\frac{2}{3} = \frac{x}{6}$; multiply both sides by 6 to solve for $x$.

Solve Equations for a Specified Variable

Isolate the specified variable using algebraic manipulation.

• Example: $T = \frac{2x}{B+E}$; solve for $x$.

Section 14.6: Problem Solving with Proportions and Rational Expressions

Use Proportions to Solve Problems

A proportion is an equation stating that two ratios are equal. The cross products property states that if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$.

• Example: If $\frac{10}{23.8} = \frac{x}{43.2}$, solve for $x$ using cross multiplication.

Solve Problems About Numbers, Work, and Distance

Rational expressions and proportions are used to solve word problems involving numbers, rates, and distances.

• Numbers: Set up equations based on relationships described in the problem.

• Work: Use rates to find time or amount of work done.

• Distance: Use $\text{Distance} = \text{Rate} \times \text{Time}$ and set up equations for unknowns.

• Example: If a car travels 280 miles and a motorcycle 240 miles in the same time, and the car's speed is 10 mph more, set up equations to solve for each speed.

Summary Table: Operations with Rational Expressions

Operation	Key Steps	Example
Simplify	Factor, cancel common factors	$\frac{x^2-9}{x^2-6x+9}$
Multiply	Multiply numerators and denominators	$\frac{3}{10} \times \frac{2x+10}{x+5}$
Divide	Multiply by reciprocal	$\frac{x+2}{7-x} \div \frac{x^2-5x+6}{7-x}$
Add/Subtract (Same Denominator)	Add/subtract numerators	$\frac{2}{3} + \frac{5}{3}$
Add/Subtract (Unlike Denominator)	Find LCD, rewrite, combine	$\frac{5x+7}{x} + \frac{4x}{x^2}$
Solve Equation	Multiply by LCD, solve, check	$\frac{2}{3} = \frac{x}{6}$

Additional info: These notes expand on the provided worksheet structure, filling in academic context and examples for clarity and completeness.