Comet builds (x + 3)(x + 3) with tiles and stares at it. "A square. One x squared, six strips, nine units."
"x² + 6x + 9," Wren says. "Now try (x + 3)(x - 3). What do you notice?"
"The strips cancel," Comet says. "Three plus, three minus. x² - 9. No middle term at all."
Nova projects a longer expression, 4x² + 12x + 9. "Would you like a hint? Is the first term a square? The last?"
"2x squared and 3 squared," Comet says. "And the middle is 2 times 2x times 3. It is (2x + 3) squared."
"Seeing the structure," Wren says. "You rewrite an expression by recognizing its shape, not by guessing."
"Our engineer," Comet says, "spot the shapes in today's list, then let us review the week."
The square of a binomial: (x + a)² = x² + 2ax + a². For a = 3, (x + 3)² = x² + 6x + 9.
The difference of squares: (x + a)(x - a) = x² - a². For a = 3, (x + 3)(x - 3) = x² - 9.
In the square, the middle term is twice the product of the two parts. In the difference, the middle terms cancel.
| Product | Expanded | Shape |
|---|---|---|
| (x + 4)² | x² + 8x + 16 | square of a binomial |
| (x - 5)² | x² - 10x + 25 | square of a binomial |
| (x + 6)(x - 6) | x² - 36 | difference of squares |
| (2x + 3)² | 4x² + 12x + 9 | square of a binomial |
| (3x + 1)(3x - 1) | 9x² - 1 | difference of squares |
An expression has a shape. 4x² + 12x + 9 is a square plus twice a product plus a square, so it is (2x + 3)².
x⁴ - 16 is (x²)² - 4², a difference of squares with x² playing the part of x.
Everyday shapes: a square frame with sides x + 2 has area (x + 2)². A border cut from a square leaves a difference of squares.
Next week you run these shapes backwards. Seeing them now makes factoring quick.
| Statement | True or false? |
|---|---|
| (x + 3)² = x² + 9 | ? |
| (x + 3)² = x² + 6x + 9 | ? |
| (x + 6)(x - 6) = x² - 36 | ? |
| (2x + 3)² = 4x² + 12x + 9 | ? |
| Adding, subtracting or multiplying polynomials always gives a polynomial. | ? |
Terrific week. You can add, subtract and multiply polynomials and see the shape of an expression. Tomorrow is Loft Day.