Overview
This lecture covers techniques for simplifying rational expressions by factoring, reducing terms, and canceling common factors in numerators and denominators.
Simplifying Basic Rational Expressions
- Simplify by dividing coefficients and subtracting exponents of like bases.
- Example: ( \frac{35x^5}{49x^2} = \frac{5x^3}{7} ) after reducing by 7 and subtracting exponents.
Factoring and Canceling in Rational Expressions
- Factor out the greatest common factor (GCF) from both numerator and denominator when possible.
- Example: ( \frac{4x^2 + 8x}{3x + 6} ) factors to ( \frac{4x(x+2)}{3(x+2)} = \frac{4x}{3} ) after canceling ( x+2 ).
Special Factoring Cases
- Use difference of squares: ( x^2 - 16 = (x+4)(x-4) ).
- Factor trinomials: Find two numbers that multiply to the last term and add to the middle coefficient.
- Example: ( \frac{x^2 - 16}{x^2 + 9x + 20} = \frac{(x+4)(x-4)}{(x+4)(x+5)} = \frac{x-4}{x+5} ).
Handling Negative Factors
- If expressions look similar but with opposite signs (e.g., ( 5-x ) vs. ( x-5 )), factor out -1 to aid cancellation.
- Example: ( \frac{5-x}{x-5} = -1 ).
Practice Problems Review
- Always factor and cancel shared terms.
- Reduce coefficients and subtract exponents for variables with the same base.
- Be alert for situations requiring factoring by grouping or use of difference of squares.
Key Terms & Definitions
- Rational Expression — A fraction with polynomials in the numerator and denominator.
- Greatest Common Factor (GCF) — The largest factor shared by all terms in an expression.
- Difference of Squares — A factorization method: ( a^2-b^2 = (a+b)(a-b) ).
- Trinomial — A polynomial with three terms.
- Factor by Grouping — A factoring technique for polynomials with four terms.
Action Items / Next Steps
- Practice simplifying rational expressions as outlined in the lecture.
- Complete assigned practice problems from the lesson.