Final answer:
The square of a binomial, whether expressed as (x + a)² or (x - a)², results in a trinomial that includes the square of the first term (x²), twice the product of both terms (2ax or -2ax), and the square of the second term (a²). These products demonstrate the application of the binomial theorem and the importance of series expansions in algebra.
Step-by-step explanation:
The student asked about finding each product for the expression (x + 9)(x + 9), (x - 7)(x - 7), and (2x - 1)² and identifying what all products of the square of a binomial have in common. To find the products of binomials being squared, we use the formula for the square of a binomial which is (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b².
Let's calculate the products:
1. (x + 9)(x + 9) = x² + 18x + 81
2. (x - 7)(x - 7) = x² - 14x + 49
3. (2x - 1)² = 4x² - 4x + 1
All these products share a common characteristic: they follow the binomial theorem in which the square of a binomial is also a trinomial with a quadratic term, a linear term, and a constant term. Specifically, each product is composed of the square of the first term, twice the product of the two terms, and the square of the second term. This common form among the products of binomials squared demonstrates an essential aspect of algebra known as series expansions.