Section 7.4 Rational Functions
Rational functions are algebraic fractions, so in this lesson we review the basic operations with algebraic fractions.
Subsection 1. Multiply and divide fractions
Subsubsection Examples
Example 7.39.
Multiply \(~\dfrac{4y^2-1}{4-y^2} \cdot \dfrac{y^2-2y}{4y+2}\)
We factor each numerator and denominator, and look for common factors.
Example 7.40.
Divide \(~\dfrac{6ab}{2a+b} \div (4a^2b)\)
We multiply the first fraction by the reciprocal of the second fraction.
Subsubsection Exercises
Checkpoint 7.41.
Multiply \(~\dfrac{3xy}{4xy-6y^2} \cdot \dfrac{2x-3y}{12x}\)
Checkpoint 7.42.
Multiply \(~\dfrac{9x^2-25}{2x-2} \cdot \dfrac{x^2-1}{6x-10}\)
Checkpoint 7.43.
Divide \(~(x^2-9) \div \dfrac{x^2-6x+9}{3x}\)
Checkpoint 7.44.
Divide \(~\dfrac{x^2-1}{x+3} \div \dfrac{x^2-x-2}{x^2+5x+6}\)
Subsection 2. Add and subtract fractions
Subsubsection Examples
Example 7.45.
Subtract \(~\dfrac{3x}{x+2} - \dfrac{2x}{x-3}\)
Step 1: The LCD for the fractions is \((x+2)(x-3)\text{.}\)
Step 2: We build each fraction to an equivalent one with the LCD.
Step 3: Combine the numerators over the same denominator.
Step 4: Reduce the result, if possible. We factor numerator and denominator to find
The fraction cannot be reduced.
Example 7.46.
Write as a single fraction \(~1+\dfrac{2}{a} - \dfrac{a^2+2}{a^2+a}\)
Step 1: By factoring each denominator, we find that the LCD for the fractions is \(a(a+1)\text{.}\)
Step 2: We build each fraction to an equivalent one with the LCD.
Step 3: Combine the numerators over the same denominator.
Step 4: Reduce the fraction to find
Subsubsection Exercises
Checkpoint 7.47.
Subtract \(~\dfrac{x+1}{x^2+2x} - \dfrac{x-1}{x^2-3x}\)
Checkpoint 7.48.
Write as a single fraction \(~y-\dfrac{2}{y^2-1} + \dfrac{3}{y+1}\)
Subsection 3. Simplify complex fractions
Subsubsection Example
Example 7.49.
Simplify \(~\dfrac{x+\dfrac{3}{4}}{x-\dfrac{1}{2}}\)
Consider all the simple fractions that appear in the complex fraction; in this case their LCD is 4. We multiply each term of the numerator and each term of the denominator by 4.
The original complex fraction is equivalent tp the simple fraction \(\dfrac{4x+3}{4x-2}\text{.}\)
Subsubsection Exercises
Checkpoint 7.50.
Write the complex fraction as a simple fraction in lowest terms:
Checkpoint 7.51.
Write the complex fraction as a simple fraction in lowest terms: