Sums, Differences, and Squares
Table 8.2 provides a summary of operations involving a few of the most common forms of binomial expressions. The discussion that follows examines a few expressions that illustrate the application of these generalized approaches to working with binomials.
| Item | Discussion |
|---|---|
| (a − b)(a + b) = a2 − b2 | The product of the sum and the difference of the same two terms consists of the square of the second term subtracted from the square of the first term. |
| (a + b)2 = a2 + 2ab + b2 | The square of the sum of two terms consists of the sum of the square of the first term, twice the product of the two terms, and the square of the second term. |
| (a − b)2 = a2 − 2ab + b2 | The square of the difference of two terms consists of the square of the first term minus twice the product of the first and second terms, plus the square of the second term. |
As an illustration of the product of the sum and difference of the same binomial, consider this expression:
| (m + 3)(m − 3) | |
| = m2 − 3m + 3m − 9 | The middle terms cancel out. |
| = m2 − 9 | The difference of the squares remains. |
Here is an example of the square of the sum of two terms:
(m + 3)2
= m (m)+ 3m + 3m + 3(3)
= m2 + 6m + 9
Finally, here is an example of the square of the difference of two terms:
(m − 3)2
= (m − 3)(m − 3)
= m (m)− 3m − 3m + 3(3)
= m2 − 6m + 9
Exercise Set 8.2Identify each expression as a sum, difference, or square (see Table 8.2).
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