Rational Expressions

What are Rational Expressions?

Rational expression: Fraction where numerator and denominator are polynomials

Examples:

• 3/x
• (x + 2)/(x - 5)
• (x² - 4)/(2x + 6)

Key rule: Denominator cannot equal zero!

Excluded Values

Excluded values: Values that make denominator = 0

Example 1: Find Excluded Values

Expression: 5/(x - 3)

Set denominator = 0:

x - 3 = 0
x = 3

Excluded value: x ≠ 3

Example 2: Quadratic Denominator

Expression: (x + 1)/(x² - 9)

Factor denominator:

x² - 9 = (x + 3)(x - 3)

Set equal to 0:

x + 3 = 0  or  x - 3 = 0
x = -3  or  x = 3

Excluded values: x ≠ -3, x ≠ 3

Simplifying Rational Expressions

Like simplifying fractions: Factor and cancel common factors

Steps:

1. Factor numerator completely
2. Factor denominator completely
3. Cancel common factors
4. State excluded values

Example 1: Simple Factoring

Simplify: (2x + 6)/(x + 3)

Factor numerator:

2x + 6 = 2(x + 3)

Simplify:

2(x + 3)/(x + 3) = 2

Excluded value: x ≠ -3

Answer: 2 (where x ≠ -3)

Example 2: Difference of Squares

Simplify: (x² - 4)/(x + 2)

Factor numerator:

x² - 4 = (x + 2)(x - 2)

Simplify:

(x + 2)(x - 2)/(x + 2) = x - 2

Excluded value: x ≠ -2

Answer: x - 2 (where x ≠ -2)

Example 3: Both Factor

Simplify: (x² - x - 6)/(x² - 9)

Factor numerator:

x² - x - 6 = (x - 3)(x + 2)

Factor denominator:

x² - 9 = (x + 3)(x - 3)

Simplify:

(x - 3)(x + 2)/[(x + 3)(x - 3)] = (x + 2)/(x + 3)

Excluded values: x ≠ 3, x ≠ -3

Answer: (x + 2)/(x + 3)

Multiplying Rational Expressions

Just like fractions: Multiply numerators, multiply denominators

Better method:

1. Factor everything
2. Cancel common factors
3. Multiply remaining factors

Example 1: Multiply and Simplify

(x/4) · (8/x²)

Multiply:

(x · 8)/(4 · x²) = 8x/(4x²)

Simplify:

8x/(4x²) = 2/x

Answer: 2/x (where x ≠ 0)

Example 2: Factor First

(x + 2)/(x - 3) · (x² - 9)/(x + 2)

Factor x² - 9:

(x + 2)/(x - 3) · [(x + 3)(x - 3)]/(x + 2)

Cancel (x + 2) and (x - 3):

= x + 3

Excluded values: x ≠ -2, x ≠ 3

Answer: x + 3

Dividing Rational Expressions

Multiply by reciprocal: a/b ÷ c/d = a/b · d/c

Example 1: Divide

(x/5) ÷ (x/10)

Multiply by reciprocal:

(x/5) · (10/x)

Simplify:

10x/(5x) = 2

Answer: 2 (where x ≠ 0)

Example 2: With Polynomials

(x² - 4)/(2x) ÷ (x + 2)/(4)

Multiply by reciprocal:

(x² - 4)/(2x) · 4/(x + 2)

Factor x² - 4:

[(x + 2)(x - 2)]/(2x) · 4/(x + 2)

Cancel (x + 2):

(x - 2)/(2x) · 4 = 4(x - 2)/(2x) = 2(x - 2)/x

Answer: 2(x - 2)/x or (2x - 4)/x

Adding and Subtracting (Same Denominator)

Same denominator: Add/subtract numerators, keep denominator

Example: Same Denominator

(3x)/(x + 5) + (2x)/(x + 5)

Add numerators:

(3x + 2x)/(x + 5) = 5x/(x + 5)

Answer: 5x/(x + 5)

Adding and Subtracting (Different Denominators)

Find LCD (Least Common Denominator):

1. Factor all denominators
2. LCD = product of all unique factors (highest powers)
3. Create equivalent fractions
4. Add/subtract numerators

Example 1: Different Denominators

2/x + 3/(2x)

LCD: 2x

Create equivalent fractions:

2/x · (2/2) + 3/(2x) = 4/(2x) + 3/(2x)

Add:

(4 + 3)/(2x) = 7/(2x)

Answer: 7/(2x)

Example 2: With Binomials

5/(x + 2) + 3/(x - 1)

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

Create equivalent fractions:

5/(x + 2) · [(x - 1)/(x - 1)] + 3/(x - 1) · [(x + 2)/(x + 2)]

Expand:

[5(x - 1)]/[(x + 2)(x - 1)] + [3(x + 2)]/[(x + 2)(x - 1)]

Simplify numerators:

(5x - 5 + 3x + 6)/[(x + 2)(x - 1)]

Combine:

(8x + 1)/[(x + 2)(x - 1)]

Answer: (8x + 1)/[(x + 2)(x - 1)]

Example 3: Subtraction

x/(x - 3) - 2/(x + 3)

LCD: (x - 3)(x + 3)

Create equivalent fractions:

[x(x + 3)]/[(x - 3)(x + 3)] - [2(x - 3)]/[(x - 3)(x + 3)]

Expand:

(x² + 3x - 2x + 6)/[(x - 3)(x + 3)]

Simplify:

(x² + x + 6)/[(x - 3)(x + 3)]

Answer: (x² + x + 6)/(x² - 9)

Complex Fractions

Complex fraction: Fraction within a fraction

Method: Multiply by LCD of all small fractions

Example: Simplify Complex Fraction

(1/x + 1/2) / (1/x - 1/3)

LCD of small fractions: 6x

Multiply numerator and denominator by 6x:

[6x(1/x + 1/2)] / [6x(1/x - 1/3)]
= (6 + 3x) / (6 - 2x)

Factor if possible:

= 3(2 + x) / [2(3 - x)]

Answer: 3(2 + x)/[2(3 - x)]

Real-World Applications

Rate problems: Distance/time, work/time

Combined rates: 1/a + 1/b = 1/t

Proportion problems: Scaling recipes, maps

Electrical resistance: Parallel circuits use reciprocals

Example: Work Rate

Person A completes job in 3 hours. Person B in 6 hours. Together?

Rate of A: 1/3 per hour Rate of B: 1/6 per hour

Combined rate:

1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2

Time together:

1 ÷ (1/2) = 2 hours

Answer: 2 hours working together

Practice