Operations with Rational Expressions: Foundations

[Video: Introduction to Rational Expressions]

Learn how to simplify, add, subtract, multiply, and divide rational expressions - fractions with polynomials in the numerator and denominator.

Understanding Rational Expressions

What are Rational Expressions?

Fractions where both numerator and denominator are polynomials.

Examples: (x+1)/(x-2), (3x²-5)/(2x+7), 5/(x³-1)

Important Restrictions

The denominator cannot equal zero. Always identify excluded values.

For (x+1)/(x-2), x ≠ 2

Simplifying Rational Expressions

Method: Factor and Cancel

Factor numerator and denominator completely
Cancel common factors
State any restrictions

Example: Simplify (x²-9)/(x²+6x+9)

1. Factor: [(x+3)(x-3)]/[(x+3)(x+3)]

2. Cancel: (x-3)/(x+3)

3. Restrictions: x ≠ -3

Operations with Rational Expressions

Multiplication and Division

Multiplication: Multiply numerators and denominators

(a/b) × (c/d) = (ac)/(bd)

Division: Multiply by the reciprocal

(a/b) ÷ (c/d) = (a/b) × (d/c) = (ad)/(bc)

Example: [(x²-4)/(x+1)] × [(x+3)/(x-2)]

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

= (x+2)(x+3)/(x+1), x ≠ 2, -1

Addition and Subtraction

Requires a common denominator:

Find the Least Common Denominator (LCD)
Rewrite each fraction with LCD
Combine numerators
Simplify

Example: (3/x) + (2/(x+1))

1. LCD: x(x+1)

2. Rewrite: [3(x+1) + 2x]/[x(x+1)]

3. Combine: (5x+3)/[x(x+1)]

Key Takeaways

Always factor first when possible
For multiplication/division, cancel common factors before multiplying
For addition/subtraction, find LCD before combining
Always state restrictions on the variable
Check your work by plugging in test values

Practice Question

Which of the following is equivalent to [1/(x-3)] - [2/(x²-9)] for all x ≠ ±3?

A) (x-1)/(x²-9)

B) (x-5)/(x²-9)

C) (x+1)/(x²-9)

D) (x+5)/(x²-9)

Follow these steps:

Factor denominators to find LCD
Rewrite each fraction with the common denominator
Combine the numerators
Simplify the result

Step-by-Step Solution:

1. Factor denominator: x²-9 = (x+3)(x-3)

2. LCD is (x+3)(x-3)

3. Rewrite first term: [1/(x-3)] = (x+3)/[(x+3)(x-3)]

4. Now subtract: [(x+3)-2]/[(x+3)(x-3)]

5. Simplify numerator: (x+1)/[(x+3)(x-3)] = (x+1)/(x²-9)

The correct answer is C) (x+1)/(x²-9).