Technique explanation
Polynomial multiplication follows the same 'vertically and crosswise' pattern as multi-digit numbers. Each column in the pattern corresponds to a specific power of x. Unlike numerical multiplication, there are no carries between columns because each term represents a different variable power.
Method Condition: Works for any two polynomials; most efficient for quadratics and cubics.
Standard fallback: The distributive property (FOIL for binomials) or grid/tabular multiplication.
Worked examples
Example 1: Two Binomials
Calculate (x + 2)(x + 3).
- Coefficients: (1, 2) and (1, 3).
- Right vertical: 2 × 3 = 6 (constant).
- Crosswise: (1 × 3) + (2 × 1) = 5 (x term).
- Left vertical: 1 × 1 = 1 (x² term).
- Result: x² + 5x + 6.
- Check: Substitute x = 1: (1 + 2)(1 + 3) = 12; 1² + 5(1) + 6 = 12 (Equality holds).
Example 2: Two Quadratics
Calculate (3x² + 2x + 1)(x² + 4x + 5).
- Coefficients: (3, 2, 1) and (1, 4, 5).
- x⁰: 1 × 5 = 5.
- x¹: (2 × 5) + (1 × 4) = 14.
- x²: (3 × 5) + (2 × 4) + (1 × 1) = 24.
- x³: (3 × 4) + (2 × 1) = 14.
- x⁴: 3 × 1 = 3.
- Result: 3x⁴ + 14x³ + 24x² + 14x + 5.
- Check: Substitute x = 1: (3+2+1)(1+4+5) = 60; 3(1)⁴ + 14(1)³ + 24(1)² + 14(1) + 5 = 60.
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