Question

1. What patterns are there in the product of the square of a binomial and the product of a sum and a difference ?

4. Explain why the product of two binomials in the form (a + b)(a - b) is a binomial instead of a trinomial.

Answer 1

1. Patterns

- Square of a binomial:
(a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2.
(a - b)^2 = (a - b)(a - b) = a^2 - 2ab + b^2.
Pattern: the first and last terms are the squares a^2 and b^2; the middle term is ±2ab (positive when +, negative when −).

Example: (x + 3)^2 = x^2 + 6x + 9.

- Product of a sum and a difference (difference of squares):
(a + b)(a - b) = a^2 - b^2.
Pattern: you get a binomial with the squares of the two terms and the middle terms cancel.

Example: (x + 3)(x - 3) = x^2 - 9.

4. Why (a + b)(a - b) is a binomial instead of a trinomial

Use FOIL (expand):
(a + b)(a - b) = a·a + a·(−b) + b·a + b·(−b)
= a^2 − ab + ab − b^2.
The −ab and +ab cancel, leaving a^2 − b^2, which has only two terms. That cancellation is why the product is a binomial (the "middle" terms are additive inverses). An intuitive area interpretation: the rectangular areas corresponding to the mixed terms are equal in magnitude but opposite in sign, so they remove each other.