Overview
This lesson covers factoring trinomials with a leading coefficient not equal to 1, using the reverse FOIL process and trial and error.
Reverse FOIL Method Overview
- FOIL stands for First, Outer, Inner, Last—method to multiply two binomials.
- To factor, write the trinomial as a product of two binomials.
- The first terms in each binomial multiply to give the leading term.
- The last terms multiply to give the constant term.
- Outer and inner terms combine to match the middle term.
Example 1: Factoring When the Leading Coefficient is Prime
- For 3x² − 19x − 14, write as (3x + __)(x + __).
- List factor pairs for the constant (−14): 1, −14; −1, 14; 2, −7; −2, 7.
- Test combinations to get a middle term of −19x.
- Correct factorization: (3x + 2)(x − 7).
Example 2: Both Coefficients Positive
- For 2x² + 23x + 63, write as (2x + __)(x + __).
- List positive factor pairs for 63: 1,63; 3,21; 7,9.
- Check pairs for a middle term of 23x.
- Correct factorization: (2x + 9)(x + 7).
Example 3: Leading Coefficient Not Prime, Common Factor
- For 40x² − 172x + 112, first factor out a 4: 4(10x² − 43x + 28).
- Factors of 10: 5x, 2x or 10x, x; factors of 28: −1,−28; −2,−14; −4,−7.
- Reject binomials with common factors.
- Correct factorization: 4(5x − 4)(2x − 7).
Example 4: Trinomials with Two Variables
- For 8x² + 34xy + 36y², first factor out a 2: 2(4x² + 17xy + 18y²).
- Try (4x + __y)(x + __y) and (2x + __y)(2x + __y).
- Use positive pairs for 18: 1,18; 2,9; 3,6.
- Correct factorization: 2(4x + 9y)(x + 2y).
Key Terms & Definitions
- FOIL — Acronym for First, Outer, Inner, Last; method to multiply binomials.
- Trinomial — A polynomial with three terms.
- Reverse FOIL — Factoring a trinomial by finding two binomials whose product matches the original.
- Leading Coefficient — The coefficient of the highest power term in a polynomial.
- Trial and Error — Systematic checking of all factor combinations to match the middle term.
Action Items / Next Steps
- Practice factoring trinomials using the reverse FOIL and trial method.
- Complete assigned homework problems on factoring given by your instructor.