Final answer:
When the signs of the terms in a binomial being squared are different, like (a - b)^2, the result is a perfect square trinomial a^2 - 2ab + b^2. The middle term is always negative and represents twice the product of the original terms due to the mixed signs.
Step-by-step explanation:
When you square a binomial with different signs, like (a - b)2, the resulting expression will have certain characteristics. First, you will square the individual terms, a2 and b2, as usual. But with the mixed sign, the middle term of the expansion will be negative, due to the rule that the product of two numbers with different signs is negative.
Using the formula (a - b)2 = a2 - 2ab + b2, you'll find that the expression will have a negative middle term, which represents twice the product of the two terms of the original binomial. This is a key characteristic of the square of a binomial where the terms have different signs. It's also important to note that this gives a perfect square trinomial, a polynomial with three terms that is the square of a binomial.
The sign in the middle term affects whether the terms will be added or subtracted in the squared result. When the signs in the original binomial are different, you are guaranteed to have a subtraction in your final squared expression, which contrasts with the result of squaring a binomial with like signs, where both terms in the squared result are added.