Wren fills a table. "Plan A, x² + 5x + 6. If x is 1, that is 12 tiles. If x is 2, 20."
"Plan B, 2x² + 3x + 1," Comet reads. "x is 1 gives 6. x is 2 gives 15."
"Both pads together is plan C, 3x² + 8x + 7," Wren says. "What do you notice at x = 2?"
"20 plus 15 is 2015," Comet says, "and C at 2 is 35. The sum works for the numbers too."
Nova hovers over the table, her light tracing the column for x = 5. "Would you like a hint? Fill the gaps before you trust the pattern."
"Our engineer," Wren says, "finish the table. Then tell us how many more tiles B needs than A when x is 4."
A polynomial becomes a function when you name it. A(x) = x² + 5x + 6 gives the tile count of plan A for any x.
A(4) means put 4 in for x: 4² + 5 × 4 + 6 = 42.
Because C = A + B as polynomials, C(x) = A(x) + B(x) for every x. The algebra promises it, and the table shows it.
The crew's table below is their own planning data, made up for the Loft.
| x (units) | A(x) = x² + 5x + 6 | B(x) = 2x² + 3x + 1 | C(x) = 3x² + 8x + 7 |
|---|---|---|---|
| 1 | 12 | 6 | 18 |
| 2 | 20 | 15 | 35 |
| 3 | 30 | 28 | 58 |
| 4 | 42 | 45 | ? |
| 5 | 56 | ? | ? |
| 6 | 72 | ? | ? |
| Statement | True or false? |
|---|---|
| A(2) + B(2) = C(2) | ? |
| A(3) = 30 | ? |
| B(4) = 42 | ? |
| One row of a table that fits proves a polynomial identity. | ? |
| One row of a table that fails disproves a polynomial identity. | ? |
Sharp table work. Tomorrow two special products, a square and a difference of squares, and how to spot them.