Overview
This lecture covers common monomial factoring, focusing on finding the greatest common factor (GCF) in algebraic expressions and using it to factor polynomials step by step.
Finding the GCF
- To factor expressions, first identify the greatest common factor (GCF) among all terms.
- For coefficients, the GCF is the highest number that divides all coefficients.
- For variables, select the variable with the lowest exponent in each term for each variable type.
Example 1: Factoring Two-Term Expressions
- Given: ( 6xy^3 + 24x^2y )
- GCF of coefficients: 6.
- GCF of variables: ( x ) (from ( x ) and ( x^2 )), and ( y ) (from ( y^3 ) and ( y )).
- Total GCF: ( 6xy ).
- Factor result: ( 6xy(y^2 + 4x) ).
Example 2: Factoring with Negative Terms
- Given: ( 36x^3y^6 - 63x^5y^7 )
- GCF of coefficients: 9.
- GCF of variables: ( x^3 ) and ( y^6 ).
- Total GCF: ( 9x^3y^6 ).
- Factor result: ( 9x^3y^6(4 - 7x^2y) ).
Example 3: Factoring Three-Term Expressions
- Given: ( 2x^2 - 4x^3 + 18x^5 )
- GCF of coefficients: 2.
- GCF of variables: ( x^2 ).
- Total GCF: ( 2x^2 ).
- Factor result: ( 2x^2(1 - 2x + 9x^3) ).
Example 4: Factoring Expressions with Common Binomial Factors
- Given: ( 4x(x + 3) - 1(x + 3) )
- Common factor is the binomial ( (x + 3) ).
- Factor result: ( (x + 3)(4x - 1) ).
Key Terms & Definitions
- Greatest Common Factor (GCF) — The largest expression that divides each term of a given algebraic expression without remainder.
- Monomial — An algebraic expression with only one term.
- Binomial — An algebraic expression with two terms.
Action Items / Next Steps
- Practice factoring more algebraic expressions by finding the GCF.
- Review class notes and prepare for upcoming exercises on polynomial factoring.