Rational Expressions and Functions

Definition of Rational Expressions

A rational expression is a fraction in which the numerator and/or the denominator are polynomials. Rational expressions are also called fractional algebraic expressions.

• Form: \( \frac{P(x)}{Q(x)} \), where \( Q(x) \neq 0 \)

• Restriction: Any value of the variable that makes the denominator zero is excluded from the domain.

• Example: \( \frac{2x-4}{x+5} \), with restriction \( x \neq -5 \)

Basic Rule of Fractions

For any polynomials \( a, b, c \), where \( b \neq 0 \) and \( c \neq 0 \):

• \( \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd} \)

• \( \frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc} \)

Simplifying Rational Expressions

Factoring and Reducing

To simplify a rational expression, factor both the numerator and denominator completely, then divide out any common factors.

• Example: \( \frac{5x+10}{2x+2} = \frac{5(x+2)}{2(x+1)} \)

• Example: \( \frac{2x^2+9x+14}{4x+4} = \frac{(2x+7)(x+2)}{4(x+1)} \)

Special Cases: Factoring Out Negatives

When factoring a negative from a polynomial, the sign of each term changes.

• Example: \( \frac{8-6x}{3-4x} = \frac{2(4-3x)}{-(4x-3)} = -2 \)

Multiplying and Dividing Rational Expressions

Multiplication

To multiply rational expressions, multiply the numerators together and the denominators together, then simplify.

• Formula: \( \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd} \)

• Example: \( \frac{x^2-49}{x^2-25} \cdot \frac{x+7}{x-5} = \frac{(x+7)(x-7)(x+7)}{(x+5)(x-5)(x-5)} \)

Division

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

• Formula: \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc} \)

• Example: \( \frac{2x^2-8x}{4x^2+4x} \div \frac{x-4}{x+1} = \frac{2x(x-4)}{4x(x+1)} \cdot \frac{x+1}{x-4} = \frac{2x}{4x} = \frac{1}{2} \)

Adding and Subtracting Rational Expressions

Finding the Least Common Denominator (LCD)

To add or subtract rational expressions, first find the LCD of the denominators.

• Factor each denominator completely.

• List all different prime factors, each raised to the highest power that appears.

• Multiply these factors to get the LCD.

• Example: For denominators 16 and 56, \( 16 = 2^4 \), \( 56 = 2^3 \times 7 \). LCD = \( 2^4 \times 7 = 112 \).

Addition and Subtraction Process

Once the LCD is found, rewrite each fraction with the LCD as the denominator, then add or subtract the numerators.

• Formula: \( \frac{a}{b} + \frac{c}{b} = \frac{a+c}{b} \)

• Formula: \( \frac{a}{b} - \frac{c}{b} = \frac{a-c}{b} \)

• Example: \( \frac{8}{(x-1)(x+3)} - \frac{12}{(x+3)(x-4)} = \frac{8(x-4)-12(x-1)}{(x-1)(x+3)(x-4)} \)

Complex Rational Expressions

Definition and Simplification Methods

A complex rational expression (complex fraction) contains a fraction in its numerator, denominator, or both. There are two main methods for simplifying:

1. Method 1: Combine the fractions in the numerator and denominator into single fractions, then divide (multiply by the reciprocal).

2. Method 2: Multiply the numerator and denominator by the LCD of all the fractions to clear denominators, then simplify.

• Example: \( \frac{\frac{2}{c} + \frac{4}{c^2}}{\frac{7}{2c}} \) can be simplified by combining and/or multiplying by the LCD.

Rational Equations

Solving Rational Equations

To solve equations involving rational expressions:

• Find the LCD of all denominators.

• Multiply both sides of the equation by the LCD to clear denominators.

• Solve the resulting equation.

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

• Example: \( \frac{1}{x} + \frac{5}{6} = \frac{2}{3} \) → Multiply both sides by 6x, solve for x, and check the solution.

Applications: Formulas and Advanced Ratio Exercises

Solving for a Variable in a Formula

Rational equations often appear in formulas. To solve for a variable, clear denominators and isolate the desired variable.

• Example: \( \frac{1}{f} = \frac{1}{a} + \frac{1}{b} \) → Solve for a.

Proportions and Similar Triangles

Proportions are equations stating that two ratios are equal. In geometry, similar triangles have proportional sides.

• Example: If \( \frac{17}{14} = \frac{23}{y} \), cross-multiply to solve for y.

Distance, Rate, and Time Problems

Distance problems use the formula \( d = rt \) (distance = rate × time). When two objects travel the same time or distance, set up equations accordingly.

• Example: If Terry drives 360 km at r+20 km/h and Kathy drives 280 km at r km/h in the same time, \( \frac{360}{r+20} = \frac{280}{r} \).

Work Problems

Work problems involve rates of work. If two people work together, their combined rate is the sum of their individual rates.

• Example: Janice can do a job in 4 hours, Marie in 3 hours. Together: \( \frac{1}{4}x + \frac{1}{3}x = 1 \), solve for x.