Overview
This lecture covers how to multiply polynomials using the distributive property and the FOIL method, with step-by-step examples for each.
Distributive Property
- The distributive property states: a(b + c) = ab + ac.
- Used when multiplying a monomial (single term) by a binomial (two terms) or trinomial (three terms).
- Example: 5(x + 6) = 5x + 30.
- Example: 2x(5x + 8) = 10x² + 16x.
- Example: -2x²(7x³ + 5x² - 12) = -14x⁵ - 10x⁴ + 24x².
- Apply laws of exponents and basic integer operations when distributing.
FOIL Method
- FOIL stands for First, Outer, Inner, Last: a method for multiplying two binomials.
- Multiply the first terms of each binomial.
- Multiply the outer terms (first of the first and last of the second).
- Multiply the inner terms (last of the first and first of the second).
- Multiply the last terms of each binomial.
- Example: (x - 3)(x + 2) = x² + 2x - 3x - 6 = x² - x - 6.
- Example: (3x + 2)(x - 1) = 3x² - 3x + 2x - 2 = 3x² - x - 2.
- Always combine like terms after using FOIL.
Key Terms & Definitions
- Monomial — a single term algebraic expression (e.g., 2x).
- Binomial — an algebraic expression with two terms (e.g., x + 2).
- Trinomial — an algebraic expression with three terms (e.g., x² + 2x + 1).
- Distributive Property — multiplying one term by each term in a sum or difference.
- FOIL Method — a technique for multiplying two binomials (First, Outer, Inner, Last).
Action Items / Next Steps
- Practice multiplying polynomials using both the distributive property and FOIL method.
- Review laws of exponents and integer operations.
- Complete any assigned exercises on multiplying polynomials.