Overview
This lecture demonstrates factoring quadratic trinomials where the leading coefficient "A" is not 1, using trial and error, and introduces upcoming methods (grouping and "Bottoms Up").
Factoring Trinomials When A ≠1
- To factor trinomials where A ≠1 and A is not a common factor, use two binomial factors.
- The first terms of each binomial come from factors of the leading term (A x²), the second terms from factors of the constant (C).
- List all factor pairs for A and C; use trial and error to arrange them in the binomials.
- Calculate inner and outer products of terms to match the middle term (B) in the trinomial.
- If the sum of the inner and outer products does not match B, switch the order or signs of the factors.
Example 1: Factoring 4x² - 4x - 15
- Factor pairs for 4x²: (2x, 2x), (4x, x).
- Factor pairs for -15: (-3, 5), (-5, 3), (-1, 15), (-15, 1).
- Try (2x - 3)(2x + 5): Inner = -6x, Outer = 10x; sum is 4x (wrong sign).
- Switch signs: (2x + 3)(2x - 5): Inner = 6x, Outer = -10x; sum is -4x (correct).
Example 2: Factoring 20x² + 19x + 3
- Factor pairs for 20x²: (5x, 4x), (10x, 2x), (20x, x).
- Factor pairs for 3: (3, 1) and since B > 0, use only positive factors.
- Try combinations: (5x + 3)(4x + 1): Inner = 4x, Outer = 15x; sum is 19x (correct).
Preview of Other Methods
- The grouping technique and "Bottoms Up" method will also be shown for these examples.
Key Terms & Definitions
- Trinomial — a polynomial with three terms, in the form Ax² + Bx + C.
- Trial and Error Method — a process of testing possible factor combinations to factor a trinomial.
- Inner and Outer Product — terms from multiplying binomial factors used to sum and match the B term.
- Binomial Factor — a two-term expression used in factoring.
Action Items / Next Steps
- Review and practice factoring trinomials by trial and error for cases where A ≠1.
- Prepare to learn the grouping technique and "Bottoms Up" method in the next session.