Un-FOILing
A simple trinomial looks like \(x^2 + bx + c\). To factor it, we want to find two numbers that:
- Multiply to give \(c\).
- Add to give \(b\).
Worked Examples
Example 1: Positive Terms
Factor: \(x^2 + 7x + 10\)
- Numbers that multiply to 10: (1,10) and (2,5).
- Which pair adds to 7? (2,5).
- Result: \((x + 2)(x + 5)\)
Example 2: Negative Middle Term
Factor: \(x^2 - 8x + 12\)
- Multiply to +12, add to -8. Both numbers must be negative.
- Pairs: (-1,-12), (-2,-6), (-3,-4).
- Which adds to -8? (-2,-6).
- Result: \((x - 2)(x - 6)\)
Example 3: Negative Last Term
Factor: \(x^2 + 2x - 15\)
- Multiply to -15, add to +2. One must be positive, one negative.
- Pairs: (-1,15), (1,-15), (-3,5), (3,-5).
- Which adds to +2? (-3,5).
- Result: \((x - 3)(x + 5)\)
The Bridge to Quantum Mechanics
In Chapter 13, we will study the Quantum Harmonic Oscillator. The equation for its energy levels involves a second-order polynomial. By "factoring" this equation into two parts (which we call Ladder Operators), we can solve for all possible energy levels of the system instantly. This trinomial factoring is the foundation for the "Algebraic Method" used by modern theoretical physicists to solve the most important problems in quantum field theory.