Special products: perfect squares and the difference of squares
The square of a binomial and the product of a sum and a difference, where the patterns come from, how to use them quickly and the mistakes to avoid.
Patterns worth memorizing
Some products come up so often that it pays to know their answers by heart. They're called special products. Using the pattern is faster than multiplying term by term, and recognizing these patterns later is the key to factoring.
Squaring a binomial
Multiply \((a + b)^2\) out the long way:
\[(a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2\]
The result is called a perfect square trinomial. The two squared terms are always positive; only the middle term carries the sign.
Since squaring wipes out a sign, \((b - a)^2\) equals \((a - b)^2\). For example, \((5 - x)^2 = (x - 5)^2 = x^2 - 10x + 25\).
Sum times difference
Now multiply the sum of two terms by their difference:
\[(a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2\]
The two middle terms cancel out.
This pattern is also a mental-math trick: \(43 \cdot 37 = (40 + 3)(40 - 3) = 1600 - 9 = 1591\).
Practice
Quick check
What is \((n - 7)^2\)?
First term squared \(n^2\), twice the product \(2 \cdot n \cdot (-7) = -14n\), last term squared \(49\).
Quick check
What is \((4x + 1)(4x - 1)\)?
Sum times difference: \((4x)^2 - 1^2 = 16x^2 - 1\).
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