Let's look at the horizontal and vertical methods side-by-side in the example below.
(x\(\small\mathsf{ ^2 }\) + 3x + 1)(x\(\small\mathsf{ ^2 }\) + 2x + 4)
Horizontal Method
x\(\small\mathsf{ ^2 }\) • x\(\small\mathsf{ ^2 }\) + x\(\small\mathsf{ ^2 }\) • 2x + x\(\small\mathsf{ ^2 }\) • 4 + 3x • x\(\small\mathsf{ ^2 }\) + 3x • 2x + 3x • 4 + x\(\small\mathsf{ ^2 }\) + 2x + 4
x\(\small\mathsf{ ^4 }\) + 2x\(\small\mathsf{ ^3 }\) + 4x\(\small\mathsf{ ^2 }\) + 3x\(\small\mathsf{ ^3 }\) + 6x\(\small\mathsf{ ^2 }\) + 12x + x\(\small\mathsf{ ^2 }\) + 2x + 4
x\(\small\mathsf{ ^4 }\) + 5x\(\small\mathsf{ ^3 }\) + 11x\(\small\mathsf{ ^2 }\) + 14x + 4
Vertical Method
| x\(\small\mathsf{ ^2 }\) | + 3x | + 1 | ||
| x\(\small\mathsf{ ^2 }\) | + 2x | + 4 | ||
| 4x\(\small\mathsf{ ^2 }\) | + 12x | + 4 | ||
| 2x\(\small\mathsf{ ^3 }\) | + 6x\(\small\mathsf{ ^2 }\) | + 2x | ||
| x\(\small\mathsf{ ^4 }\) | + 3x\(\small\mathsf{ ^3 }\) | + x\(\small\mathsf{ ^2 }\) | ||
| x\(\small\mathsf{ ^4 }\) | + 5x\(\small\mathsf{ ^3 }\) | + 11x\(\small\mathsf{ ^2 }\) | + 14x | + 4 |
Do you see how easy the vertical method is? It looks a lot like a basic multiplication problem. That's why it is considered to be a shortcut for multiplying polynomials.
