A number in front of the square
A cycle shed has area 3x² + 11x + 6 square metres. Here x is a length in metres. No pair adds to 11 and multiplies to 6. So what now? Work the three moves.
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- A rectangle, sides x + 3 and 3x + 2. Here x = 4.
- Corner: 3x² = 48.
- Thin strip: 2x = 8.
- Side strip: 9x = 36.
- Corner: 3 × 2 = 6.
- Total: 48 + 8 + 36 + 6 = 98. So 3x² + 11x + 6 = (x + 3)(3x + 2).
The rule
Now the pair multiplies to the first number times the last. Split the middle term with that pair. Group the four terms in twos. Take out the shared bracket.
In 3x² + 11x + 6 the first number is 3, in front of x². The last number is 6. The product is 3 × 6 = 18. The pair adds to 11 and multiplies to 18. That pair is 2 and 9. Check: the split terms 2x and 9x multiply to 18x². And 3x² × 6 = 18x².
3x² + 2x + 9x + 6. Group in twos: x(3x + 2) + 3(3x + 2). Both halves share (3x + 2). Take it out. What is left is (x + 3). So the shed is (x + 3)(3x + 2). Check at x = 1: both forms give 20.
| Start | First × last | Pair | Split into | Brackets |
|---|---|---|---|---|
3x² + 10x + 8 | 24 | 4 and 6 | 3x² + 4x + 6x + 8 | (x + 2)(3x + 4) |
6x² + 11x + 3 | 18 | 2 and 9 | 6x² + 2x + 9x + 3 | (2x + 3)(3x + 1) |
4x² + 8x + 3 | 12 | 2 and 6 | 4x² + 2x + 6x + 3 | (2x + 3)(2x + 1) |