Special products of binomials

A binomial is a two-term expression like $x + 3$.
Multiplying binomials can be slow, but three special products show up again and again.
Learning them saves time on both expanding and factoring.

The first is the square of a sum: $(a + b)^2 = a^2 + 2ab + b^2$.
Picture a square with side $a + b$.
It splits into an $a^2$ square, a $b^2$ square, and two $ab$ rectangles.
Adding the pieces gives $a^2 + 2ab + b^2$, so notice the middle term is $2ab$, not just $ab$.

The square of a difference is almost the same: $(a - b)^2 = a^2 - 2ab + b^2$.
The only change is the middle sign, which is now negative.
For example, $(x - 4)^2 = x^2 - 8x + 16$.

The third pattern is the difference of squares: $(a + b)(a - b) = a^2 - b^2$.
The two middle terms cancel, so no $x$-term is left.
For example, $(x + 6)(x - 6) = x^2 - 36$.

These patterns appear all over the SAT.
The equation of a circle uses $(x - a)^2$ and $(y - b)^2$.
And spotting $a^2 - b^2$ lets you factor it straight into $(a + b)(a - b)$.

Expand $(x + 5)^2$.
Use $(a + b)^2 = a^2 + 2ab + b^2$ with $a = x$ and $b = 5$.
So $(x + 5)^2 = x^2 + 2(x)(5) + 25 = x^2 + 10x + 25$.

Expand $(2x - 3)^2$.
Use $(a - b)^2 = a^2 - 2ab + b^2$ with $a = 2x$ and $b = 3$.
So $(2x)^2 - 2(2x)(3) + 3^2 = 4x^2 - 12x + 9$.

Expand $(x + 4)(x - 4)$.
This is a difference of squares, $(a + b)(a - b) = a^2 - b^2$, with $a = x$ and $b = 4$.
So $(x + 4)(x - 4) = x^2 - 16$.