Bigger tiles, more tiles
Identity for (px + a)(qx + b)
Two x-strips on a side
Now take a rectangle with sides 2x + 3 and 3x + 1. Each side has more than one x-strip. The area is (2x + 3)(3x + 1).
How many x²-tiles does this rectangle hold?
What this lesson covers
The idea
(px + a)(qx + b) = pqx² + (pb + aq)x + ab, which can be verified using the distributive property.
Two x-strips on a side
Now take a rectangle with sides 2x + 3 and 3x + 1. Each side has more than one x-strip. The area is (2x + 3)(3x + 1).
How many x²-tiles does this rectangle hold?
- 5
- 6
- 3
Count the tiles
Change p, q, a and b in the rectangle with sides px + a and qx + b. Count the three kinds of tile. Try three different settings.
Count with letters
(px + a)(qx + b) = pqx² + (pb + aq)x + ab, which can be verified using the distributive property.
(px + a)(qx + b) = px · qx + px · b + a · qx + a · b = pqx² + pbx + aqx + ab. For (2x + 3)(3x + 1): 2 × 3 = 6 x²-tiles, 2 × 1 + 3 × 3 = 11 x-tiles and 3 × 1 = 3 unit tiles: 6x² + 11x + 3.
Notes
(px + a)(qx + b) = pqx² + (pb + aq)x + ab: x²-tiles p × q, unit tiles a × b, and two kinds of x-strips, pb and aq.
Check yourself
In (2x + 3)(3x + 1), how many x-tiles are there?
Answer: 11
2 × 1 + 3 × 3 = 2 + 9 = 11 x-tiles.
In (4x + 1)(2x + 3), how many x²-tiles are there?
Answer: 8
4 × 2 = 8 x²-tiles.
(x + 2)(3x + 5) = 3x² + ?x + 10. What number goes in the box?
Answer: 11
1 × 5 + 2 × 3 = 5 + 6 = 11.
Which is (5x + 2)(x + 3)?
- 5x² + 7x + 6. 5 + 2 = 7 adds p and a. The x-term is pb + aq = 5 × 3 + 2 × 1 = 17.
- 5x² + 17x + 6 — correct. Yes: 5 × 1 = 5, 5 × 3 + 2 × 1 = 17, and 2 × 3 = 6.
- 6x² + 17x + 6. The x²-term is p × q = 5 × 1 = 5, not 5 + 1.