Algebraic Products
Using Vertical & Crosswise for Algebra.
Core Concepts
Did you know that the same Vertical & Crosswise method we used for numbers also works for Algebra? You can multiply expressions like (x + 2)(x + 3) in one single line!
The Pattern
For (ax + b)(cx + d):
1. Vertical (Left): Multiply the first terms (ax · cx).
2. Crosswise (Middle): Cross-multiply and add (adx + bcx).
3. Vertical (Right): Multiply the last terms (b · d).
Solved Examples
Example 1: (x + 2)(x + 3)
Question: Expand (x + 2)(x + 3).
• Step 1 (Left): x · x = x².
• Step 2 (Cross): (x·3) + (x·2) = 3x + 2x = 5x.
• Step 3 (Right): 2 · 3 = 6.
• Answer: x² + 5x + 6.
Example 2: (x + 4)(x + 5)
Question: Expand (x + 4)(x + 5).
• Step 1: x².
• Step 2: 5x + 4x = 9x.
• Step 3: 4 · 5 = 20.
• Answer: x² + 9x + 20.
Example 3: (2x + 1)(x + 3)
Question: Expand (2x + 1)(x + 3).
• Step 1: 2x · x = 2x².
• Step 2: (2x·3) + (1·x) = 6x + 1x = 7x.
• Step 3: 1 · 3 = 3.
• Answer: 2x² + 7x + 3.
Example 4: (x + 1)(x + 1)
Question: Expand (x + 1)(x + 1).
• Step 1: x².
• Step 2: 1x + 1x = 2x.
• Step 3: 1 · 1 = 1.
• Answer: x² + 2x + 1.
