Wren pins four cards to the wall. "3x² + 6x. 2x² + 7x + 3. x² - 9. x² + 6x + 9. Four shapes, four moves."
"The first has no units," Comet says. "Every term has 3x in it. Pull it out: 3x(x + 2)."
"Common factor first, always," Wren says. "The second starts with 2x squared. What do you notice?"
"Two x squared tiles. The sides are not both x plus something." Comet frowns. "One side must start with 2x."
Nova projects the middle term split in two: 6x + x. "Would you like a hint? Group the four terms in pairs."
"2x(x + 3) plus 1(x + 3). Both have x + 3. So (x + 3)(2x + 1)," Comet says. "That is sneaky."
"And the last two are last week's shapes run backwards," Wren says. "Our engineer, factor all four, then multiply back."
Look for a number and a power of x that divide every term. In 3x² + 6x, every term has 3x, so it is 3x(x + 2).
Sometimes only a number comes out: 2x² - 8 = 2(x² - 4). Then x² - 4 factors again as (x - 2)(x + 2).
Always do this move first. The leftover polynomial is smaller and easier to factor.
When the polynomial is x² + bx + c, use Monday's search. Take x² - x - 6. Two numbers multiply to -6 and add to -1: -3 and 2. So it is (x - 3)(x + 2).
When it starts with ax² and a is not 1, split the middle. For 2x² + 7x + 3, multiply a × c = 6. Find two numbers that multiply to 6 and add to 7: 6 and 1.
Rewrite 7x as 6x + x and group: 2x(x + 3) + 1(x + 3). Pull out the common binomial: (x + 3)(2x + 1).
| Shape | Pattern | Example | Factored form |
|---|---|---|---|
| difference of squares | x² - a² = (x - a)(x + a) | x² - 9 | (x - 3)(x + 3) |
| perfect square | x² + 2ax + a² = (x + a)² | x² + 6x + 9 | (x + 3)² |
| perfect square | x² - 2ax + a² = (x - a)² | x² - 8x + 16 | (x - 4)² |
| difference of squares | (2x)² - a² = (2x - a)(2x + a) | 4x² - 25 | (2x - 5)(2x + 5) |
For 2x² + 7x + 3, Comet tries tile rectangles: two x² tiles, seven strips, three units. After some sliding, the sides read 2x + 1 and x + 3.
Wren splits the middle on paper and reaches the same factors without any tiles.
Tiles show why the factors work. Splitting the middle works fast when the numbers get big.
| Statement | True or false? |
|---|---|
| 3x² + 6x = 3x(x + 2) | ? |
| x² - 9 = (x - 3)(x + 3) | ? |
| x² - 9 = (x - 3)² | ? |
| 2x² + 7x + 3 = (2x + 1)(x + 3) | ? |
| Pulling out the greatest common factor is the first move in any factoring. | ? |
Excellent sorting. Tomorrow the Loft Lab hands you bags of loose tiles, some of which refuse to make a rectangle.