Two kinds of binomial products come up so often that their patterns are worth knowing by sight. They are ordinary binomial multiplications with a shortcut.
Why it works
Square a sum: $$(a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2$$ In the area model the big square splits into an $a$-by-$a$ square, a $b$-by-$b$ square, and two equal $a$-by-$b$ rectangles; that is where the $2ab$ comes from. Multiply a sum by a difference: $(a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2$; the two middle products are opposites and cancel.
| Pattern | Example |
|---|---|
| $(a + b)^2 = a^2 + 2ab + b^2$ | $(x + 5)^2 = x^2 + 10x + 25$ |
| $(a - b)^2 = a^2 - 2ab + b^2$ | $(3x - 2)^2 = 9x^2 - 12x + 4$ |
| $(a + b)(a - b) = a^2 - b^2$ | $(2x + 7)(2x - 7) = 4x^2 - 49$ |
Key idea
$(a \pm b)^2 = a^2 \pm 2ab + b^2$ and $(a + b)(a - b) = a^2 - b^2$.
Worked examples
Worked example: Square of a difference
Expand $(3x - 2)^2$.
- $a = 3x$ and $b = 2$: $a^2 = 9x^2$, $2ab = 12x$, $b^2 = 4$.
- With the minus sign: $9x^2 - 12x + 4$.
Answer: $9x^2 - 12x + 4$
Worked example: Mental arithmetic
Use a special product to find $41 \cdot 39$.
- $41 \cdot 39 = (40 + 1)(40 - 1) = 40^2 - 1^2$.
- $1600 - 1 = 1599$.
Answer: $1599$
Common mistake
Squaring each term separately: $(x + 5)^2 = x^2 + 25$. That misses the two middle rectangles, $2 \cdot 5x = 10x$. Check with $x = 1$: $(1 + 5)^2 = 36$, while $1 + 25 = 26$; adding $10x = 10$ makes up the difference.