Rational Expressions

Rational Expressions

Rational expressions are fractions where both the numerator and denominator are polynomials. Here's what you need to know:

Basic Concepts

Definition

A rational expression has the form:

$$\frac{P(x)}{Q(x)}$$

where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

Domain

The domain of a rational expression excludes values that make the denominator zero.

Example: For $$\frac{x+3}{x^2-4}$$, the domain is all real numbers except x = 2 and x = -2 (since x² - 4 = (x+2)(x-2)).

Operations with Rational Expressions

Simplifying Rational Expressions

1. Factor both the numerator and denominator completely
2. Cancel common factors
3. Write the simplified expression

Example: Simplify $$\frac{x^2-9}{x^2-6x+9}$$

Step 1: Factor $$\frac{(x+3)(x-3)}{(x-3)^2}$$

Step 2: Cancel common factor (x-3) $$\frac{(x+3)\cancel{(x-3)}}{\cancel{(x-3)}(x-3)}$$

Step 3: Simplified expression: $$\frac{x+3}{x-3}$$ for x ≠ 3

Multiplication and Division

Multiplication: Multiply numerators and denominators

$$\frac{P(x)}{Q(x)} \cdot \frac{R(x)}{S(x)} = \frac{P(x) \cdot R(x)}{Q(x) \cdot S(x)}$$

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

Factoring: $$\frac{(x+2)(x+1)(x-1)}{(x-1)(x+2)(x-2)}$$

Canceling: $$\frac{(x+1)\cancel{(x+2)}\cancel{(x-1)}}{\cancel{(x-1)}\cancel{(x+2)}(x-2)}$$

Simplified: $$\frac{x+1}{x-2}$$ for x ≠ 1, x ≠ -2

Division: Multiply by the reciprocal of the divisor

$$\frac{P(x)}{Q(x)} \div \frac{R(x)}{S(x)} = \frac{P(x)}{Q(x)} \cdot \frac{S(x)}{R(x)} = \frac{P(x) \cdot S(x)}{Q(x) \cdot R(x)}$$

Example: $$\frac{x^2-4}{x+1} \div \frac{x-2}{x^2-1}$$

Multiply by reciprocal: $$\frac{x^2-4}{x+1} \cdot \frac{x^2-1}{x-2}$$

Factoring: $$\frac{(x+2)(x-2)(x+1)(x-1)}{(x+1)(x-2)}$$

Canceling: $$\frac{(x+2)\cancel{(x-2)}(x+1)(x-1)}{\cancel{(x+1)}\cancel{(x-2)}}$$

Simplified: $$(x+2)(x-1) = x^2+x-2$$ for x ≠ -1, x ≠ 2

Addition and Subtraction

For rational expressions with the same denominator:

$$\frac{P(x)}{R(x)} \pm \frac{Q(x)}{R(x)} = \frac{P(x) \pm Q(x)}{R(x)}$$

For rational expressions with different denominators:

1. Find the least common denominator (LCD)
2. Rewrite each fraction with the LCD
3. Add or subtract the numerators
4. Simplify if possible

Example: $$\frac{3}{x-2} + \frac{4}{x+1}$$

LCD = (x-2)(x+1)

$$\frac{3}{x-2} + \frac{4}{x+1} = \frac{3(x+1)}{(x-2)(x+1)} + \frac{4(x-2)}{(x+1)(x-2)}$$

$$= \frac{3(x+1) + 4(x-2)}{(x-2)(x+1)}$$

$$= \frac{3x+3+4x-8}{(x-2)(x+1)}$$

$$= \frac{7x-5}{(x-2)(x+1)}$$

Complex Fractions

A complex fraction is a fraction where the numerator and/or denominator contains one or more fractions.

Method 1: Simplify numerator and denominator separately, then divide

Example: $$\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}$$

Numerator: $$\frac{1}{x} + \frac{1}{y} = \frac{y+x}{xy}$$

Denominator: $$\frac{1}{x} - \frac{1}{y} = \frac{y-x}{xy}$$

Divide: $$\frac{\frac{y+x}{xy}}{\frac{y-x}{xy}} = \frac{y+x}{xy} \cdot \frac{xy}{y-x} = \frac{y+x}{y-x}$$

Method 2: Multiply numerator and denominator by the LCD of all fractions

Example: $$\frac{\frac{2}{x} - \frac{1}{y}}{\frac{3}{x} + \frac{4}{y}}$$

LCD of all fractions = xy

$$\frac{\frac{2}{x} - \frac{1}{y}}{\frac{3}{x} + \frac{4}{y}} = \frac{xy(\frac{2}{x} - \frac{1}{y})}{xy(\frac{3}{x} + \frac{4}{y})}$$

$$= \frac{xy \cdot \frac{2}{x} - xy \cdot \frac{1}{y}}{xy \cdot \frac{3}{x} + xy \cdot \frac{4}{y}}$$

$$= \frac{2y - x}{3y + 4x}$$

Solving Rational Equations

A rational equation is an equation containing one or more rational expressions.

Steps to Solve

1. Find the LCD of all fractions in the equation
2. Multiply all terms by the LCD to clear fractions
3. Solve the resulting polynomial equation
4. Check all solutions in the original equation (exclude values that make any denominator zero)

Example: Solve $$\frac{3}{x-1} - \frac{2}{x+2} = \frac{1}{x-1}$$

LCD = (x-1)(x+2)

Multiply all terms by LCD:

$$(x-1)(x+2) \cdot \frac{3}{x-1} - (x-1)(x+2) \cdot \frac{2}{x+2} = (x-1)(x+2) \cdot \frac{1}{x-1}$$

$$3(x+2) - 2(x-1) = (x+2)$$

$$3x+6 - 2x+2 = x+2$$

$$x+8 = x+2$$

$$8 = 2$$

This is a contradiction, so there is no solution.

Applications

Work Problems

If person A can complete a task in a hours and person B can complete the same task in b hours, then working together they can complete the task in $$\frac{ab}{a+b}$$ hours.

Example: If John can paint a room in 4 hours and Mary can paint the same room in 6 hours, how long will it take them to paint the room working together?

$$\frac{4 \cdot 6}{4+6} = \frac{24}{10} = 2.4$$ hours

Motion Problems

Distance = Rate × Time, or Time = Distance ÷ Rate

Example: A boat travels 24 miles downstream in 2 hours and 24 miles upstream in 6 hours. Find the speed of the boat in still water and the speed of the current.

Let b = boat speed in still water and c = current speed

Downstream: Rate = b + c, Time = 24 ÷ (b + c) = 2

Upstream: Rate = b - c, Time = 24 ÷ (b - c) = 6

From first equation: b + c = 12

From second equation: b - c = 4

Solving: b = 8 mph, c = 4 mph

Common Pitfalls and Tips

• Always check for domain restrictions (values that make denominators zero)
• When simplifying, factor completely before canceling
• Remember that $$\frac{a+b}{c} ≠ \frac{a}{c} + \frac{b}{c}$$ but $$\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}$$
• When solving rational equations, always check your solutions in the original equation
• Be careful with signs when combining fractions with different denominators