Overview
This lecture covers techniques for factoring polynomials, including identifying the greatest common factor, using common formulas, and applying factoring strategies for quadratic and higher-degree expressions.
Factoring Basics
- Factoring reverses distribution: a(b + c) = ab + ac.
- The greatest common factor (GCF) is the largest factor shared by all terms; pull it out to begin factoring.
- Divide each term by the GCF to determine remaining factors inside the parentheses.
Common Factoring Formulas
- Difference of squares: a² - b² = (a - b)(a + b).
- Perfect square trinomial: (a ± b)² = a² ± 2ab + b².
- Sum of squares: a² + b² is prime and cannot be factored over real numbers.
Factoring Quadratic Expressions
- Quadratics in the form ax ² + bx + c require finding two numbers that multiply to c and add or subtract to b.
- If c is negative, the two factors have opposite signs; if c is positive, both signs match b’s sign.
- Select factors so the larger matches the sign of b.
- Verify the factorization by multiplying it back out (FOIL method): First, Outer, Inner, Last.
Factoring When Leading Coefficient (a) ≠ 1
- Multiply the leading coefficient by c to find possible factor pairs.
- Find two numbers that multiply to ac and sum (or subtract) to b.
- Break the middle term and factor by grouping if necessary.
Factoring Higher Degree and Grouping
- Always check first for a GCF among all terms.
- Factor higher powers like x⁴ - 16 as (x² + 4)(x² - 4) using difference of squares.
- Continue factoring any differences of squares until all terms are prime.
- For four-term polynomials, factor by grouping: pull out the GCF from each pair and factor common binomial factors.
Key Terms & Definitions
- Factoring — rewriting an expression as a product of simpler factors.
- Greatest Common Factor (GCF) — largest factor that divides all terms.
- Difference of squares — a² - b² = (a - b)(a + b).
- Prime polynomial — a polynomial that cannot be factored over the real numbers.
Action Items / Next Steps
- Practice factoring quadratic expressions using the discussed strategies.
- Review homework on factoring by GCF, difference of squares, and grouping.
- Double-check your factorizations by multiplying them back out.