Overview
This lecture explains how to find the Greatest Common Factor (GCF) or greatest common monomial factor in algebra, including factoring polynomials using the GCF and the distributive property.
Greatest Common Factor (GCF)
- The GCF of numbers or terms is the largest factor they share.
- GCF can be found by listing all factors or by using prime factorization.
- For variables, the GCF uses the variable with the lowest exponent present in all terms.
Prime Factorization
- Break numbers into products of prime numbers (e.g., 6 = 2 × 3).
- For monomials, include variable parts with their exponents.
- In finding GCF, use only primes and the lowest exponent for common variables.
Monomials and Polynomials
- A monomial has one term (e.g., 7x, -3y²).
- A polynomial is an expression combining constants and variables by addition or subtraction.
- The standard form of a polynomial lists terms in descending order of degree.
Finding GCF of Monomials (Examples)
- Example: GCF of 6x² and 15x⁴ is 3x².
- Example: GCF of 6a and 18ab is 6a.
- Example: GCF of 10a and 12a²b is 2a.
- Example: GCF of -8x²y and 16xy is 8xy.
Factoring Polynomials Using GCF
- Rewrite the polynomial as a product: GCF × (remaining polynomial).
- Divide each term by the GCF for the terms inside parentheses.
- Example: 4x² + 6x = 2x(2x + 3).
- Example: 3x² + 6x = 3x(x + 2).
- Use the distributive property to factor out the GCF.
Factoring with Binomials as GCF
- Look for common binomial factors (e.g., (a + 3) in multiple terms).
- Factor complex polynomials by rearranging and finding GCF of expressions.
Key Terms & Definitions
- GCF (Greatest Common Factor) — The largest factor common to each term in an expression.
- Prime Factorization — Expressing a number as a product of prime numbers.
- Monomial — An algebraic expression with only one term.
- Polynomial — An algebraic expression with one or more terms.
- Standard Form — Writing a polynomial with terms in descending order of degree.
- Distributive Property — a(b + c) = ab + ac, used to factor expressions.
Action Items / Next Steps
- Practice finding the GCF of given sets of monomials.
- Factor polynomials completely by extracting the GCF and using the distributive property.
- Arrange polynomials in standard form before factoring.