Overview
This lecture covers finding the Greatest Common Factor (GCF) of numbers and monomials, and using the GCF to factor polynomials by applying the distributive property.
Understanding GCF (Greatest Common Factor)
- The GCF is the largest factor shared by two or more terms.
- For numbers, list factors or use prime factorization to find the GCF.
- Prime factors are numbers only divisible by 1 and themselves.
- For monomials, the GCF includes the greatest shared numerical factor and each variable with the smallest exponent present in all terms.
Key Terms: Monomials, Polynomials, and Standard Form
- A monomial consists of a single term (mono = one).
- A polynomial is an expression with multiple terms involving constants and variables.
- Exponents for variables must be whole numbers (no negatives or fractions).
- Standard form arranges polynomial terms in descending order of exponents.
Finding GCF by Listing and Prime Factorization
- Example: GCF of 6x² and 15x⁴ is found by listing or factorizing (GCF: 3x²).
- Factor numbers into primes, and for variables, choose the lowest exponent common to all terms.
- Apply the process step-by-step for each pair of monomials.
Example GCF Calculations
- GCF of 6a and 18ab is 6a.
- GCF of 10a and 12a²b is 2a.
- GCF of -8x²y and 16xy is 8xy.
- GCF of 8ab³ and 10a²b² is 2ab².
Factoring Polynomials Using GCF and Distributive Property
- Factoring means rewriting a polynomial as a product using the GCF.
- Example: 4x² + 6x factors as 2x(2x + 3).
- Divide each term by the GCF, and place the result inside parentheses.
- Use the distributive property to verify factored form.
- For variable exponents, subtract exponents when dividing like bases.
Special Cases: Binomial and Polynomial GCFs
- Sometimes the GCF is a binomial (e.g., a + 3 or 3b - 1).
- Rearrange terms if needed to identify and factor out common binomial factors.
- When terms contain expressions like 1 - 3b and 3b - 1, adjust signs as necessary to factor efficiently.
Key Terms & Definitions
- Greatest Common Factor (GCF) — Largest shared factor between terms, using lowest exponent for shared variables.
- Prime Factor — Number only divisible by 1 and itself.
- Monomial — An algebraic expression with only one term.
- Polynomial — An expression with multiple terms (constants/variables).
- Standard Form — Order terms from highest to lowest exponent.
Action Items / Next Steps
- Practice finding GCFs of monomials and polynomials using both listing and prime factorization.
- Complete homework exercises on factoring polynomials using the GCF and distributive property.