Overview
This lecture covers how to factor trinomials by always starting with the Greatest Common Factor (GCF), then applying trial and error factoring methods.
Factoring Trinomials with the GCF
- Always factor out the GCF first before factoring the rest of the trinomial.
- The GCF is the largest expression that divides all terms in the trinomial.
- After extracting the GCF, factor the remaining trinomial using trial and error or reverse FOIL (First, Outside, Inside, Last).
Example 1: Factoring with a GCF
- For 18x^4 - 21x^3 - 15x^2, the GCF is 3x^2.
- Divide each term by 3x^2: 6x^2 - 7x - 5.
- Factor the resulting trinomial by trial and error: (2x + 1)(3x - 5).
- The fully factored form: 3x^2(2x + 1)(3x - 5).
Example 2: Factoring with Variables and GCF
- For 16x^3y - 28x^2y + 30xy^2, the GCF is 2x.
- Divide each term by 2x: 8x^2y - 14xy + 15y^2.
- Factor the trinomial: try combinations like (4x y - 3y)(2x - 5).
- Adjust signs so the middle term matches; the correct factoring: 2x(4x y - 3y)(2x - 5).
Key Terms & Definitions
- Trinomial — a polynomial with three terms.
- Greatest Common Factor (GCF) — the largest factor shared by all terms in a polynomial.
- Trial and Error Factoring — method of finding two binomials whose product equals the given trinomial.
- FOIL — method to multiply two binomials (First, Outside, Inside, Last).
Action Items / Next Steps
- Practice factoring trinomials by first finding and extracting the GCF, then factoring the remaining expression.
- Prepare for homework problems involving trinomials with numerical and variable GCFs.