Suppose 7x is split as 2x + 5x; can a similar rectangular arrangement be formed? Consider other possibilities and check.
No, a similar rectangular arrangement cannot be formed if 7x is split as 2x + 5x.
To form a rectangle using algebra tiles for x² + 7x + 12, the 7x must be split such that the two parts correspond to the dimensions of the 12 unit tiles. If we split 7x as 2x + 5x, the 12 unit tiles would need to be arranged in a 2 by 5 array. Since = 10 (not 12), the unit tiles cannot fill the space, and a complete rectangle cannot be formed.
For other possibilities, we need two numbers that add up to 7 and multiply to 12. The pairs that add up to 7 are (1, 6), (2, 5), and (3, 4). Only the pair (3, 4) has a product of 12. Thus, splitting 7x as 3x + 4x is the only possibility that allows a similar rectangular arrangement.
Explanation
The textbook context explains that for x² + 7x + 12, the 7x is split as 3x + 4x because the 12 unit tiles form a 3 by 4 array, which corresponds to the dimensions x+3 and x+4. If 7x is split as 2x + 5x, the unit tiles would require a 2 by 5 array, which only holds 10 tiles, leaving a gap. Therefore, a rectangle cannot be formed. The only successful split corresponds to factors of 12 that sum to 7.
Solution Steps
Step 1: Identify the requirement for a rectangular arrangement: the split of the x-term must match the dimensions of the unit tile array.
Step 2: Check the split 2x + 5x. This requires a 2 by 5 array for the unit tiles, which holds 10 tiles. Since we have 12 unit tiles, this arrangement is impossible.
Step 3: Consider other possibilities by finding two numbers that add to 7 and multiply to 12. The pairs are (1,6), (2,5), and (3,4).
Step 4: Conclude that only the pair (3,4) works because = 12, making 3x + 4x the only valid split.