Overview
This lecture covers methods for factoring polynomials and working with rational expressions, focusing on key techniques and definitions essential for algebra.
Factoring Polynomials
- Factoring rewrites a polynomial as a product of simpler polynomials.
- Common factoring methods include factoring out the greatest common factor (GCF), grouping, and factoring trinomials.
- When factoring by grouping, terms are grouped to factor out common factors in pairs.
- Special forms include the difference of squares: (a^2 - b^2 = (a - b)(a + b)).
- Perfect square trinomials take the form (a^2 + 2ab + b^2 = (a + b)^2).
- The zero product property states if (ab = 0), then (a = 0) or (b = 0).
Rational Expressions
- A rational expression is a ratio of two polynomials, similar to a fraction.
- Simplifying rational expressions involves factoring numerators and denominators and cancelling common factors.
- Restrictions exist where the denominator equals zero; these must be noted as excluded values.
- To multiply rational expressions, multiply numerators and denominators, then simplify.
- To divide, multiply by the reciprocal of the divisor and simplify.
- Adding or subtracting rational expressions requires a common denominator.
Key Terms & Definitions
- Polynomial — an expression with terms added or subtracted, each term being a constant times a variable raised to a whole number power.
- Greatest Common Factor (GCF) — the largest expression that divides each term of a polynomial.
- Trinomial — a polynomial with exactly three terms.
- Difference of squares — a binomial in the form (a^2 - b^2).
- Rational expression — a fraction where the numerator and/or denominator are polynomials.
Action Items / Next Steps
- Practice factoring different types of polynomials, including trinomials and special products.
- Complete assigned homework problems on simplifying and operating with rational expressions.
- Learn to identify and state restrictions on the variable for rational expressions.