Overview
This lecture reviews how to add and subtract similar fractions and focuses on adding and subtracting rational algebraic expressions with like denominators.
Addition and Subtraction of Similar Fractions
- To add or subtract similar fractions, add or subtract the numerators and copy the common denominator.
- After combining, simplify the result if possible by reducing to lowest terms.
- Example: ( \frac{2}{4} + \frac{1}{4} = \frac{3}{4} ).
- Example: ( \frac{4}{6} - \frac{2}{6} = \frac{2}{6} = \frac{1}{3} ) after simplification.
Adding and Subtracting Rational Algebraic Expressions
- With like denominators, add or subtract the numerators and copy the common denominator.
- Factor the numerator, if possible, and cancel any common factors with the denominator.
- Example: ( \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c} ); ( \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c} ), where ( c \neq 0 ).
Worked Examples
- ( \frac{2a}{4b} + \frac{3}{4b} = \frac{2a+3}{4b} ) (no factoring possible).
- ( \frac{10c}{13} + \frac{16c}{13} = \frac{26c}{13} = 2c ) after factoring and cancelling 13.
- ( \frac{8d-3}{9} + \frac{4d+12}{9} = \frac{12d+9}{9} = \frac{4d+3}{3} ) after combining like terms and simplifying.
Subtracting Rational Expressions: Special Cases
- Change the signs of all terms in the subtrahend before subtracting.
- For difference of squares: ( \frac{9x^2}{3x-5} - \frac{25}{3x-5} = \frac{9x^2-25}{3x-5} = 3x+5 ) after factoring and cancelling.
- For expressions with common factors, factor, cancel, and simplify the answer.
Key Terms & Definitions
- Similar fractions — Fractions with the same denominator.
- Rational algebraic expressions — Fractions whose numerators and denominators are polynomials.
- Like denominators — Fractions or expressions sharing the same denominator.
- Difference of squares — An expression ( a^2 - b^2 ) factoring as ( (a-b)(a+b) ).
Action Items / Next Steps
- Practice adding and subtracting rational algebraic expressions with like denominators.
- Simplify answers by factoring and reducing to lowest terms where possible.