Special Products

Binomial squares and the cross term you must not drop.

The idea

The special products are products that occur so often that we learn their expansions as patterns rather than multiplying them out every time. The first to learn is the square of a binomial.

Proposition (Square of a binomial).

For all numbers $x$ and $a$, $(x+a)^{2} = x^{2} + 2ax + a^{2} \qquad\text{and}\qquad (x-a)^{2} = x^{2} - 2ax + a^{2}.$

The exponent in $(x+a)^{2}$ belongs to the whole parenthesis, so the expression means $(x+a)(x+a)$, an ordinary product of two factors. Multiplying it out, the two $x$ terms give $x^{2}$, the two constants give $a^{2}$, and the two mixed pairings each give $ax$, so they add. The middle term $2ax$ is the cross term, and dropping it — writing only $x^{2} + a^{2}$ — is the most common error with this pattern. Drawn as the area of a square with side $x + a$, the identity is visible: an $x^{2}$ square, two $ax$ rectangles, and an $a^{2}$ corner tile it exactly.

A subtraction behaves the same way with one sign flipped: each mixed pairing now carries a minus sign, so the middle term is negative, while the constant $(-a)(-a) = a^{2}$ stays positive.

Ways to work on it

• Walkthrough. Square positive and negative binomials, then catch the missing-cross-term mistake.
• Practice. Expand a random binomial square.
• Hardest. A binomial square with a leading coefficient.

Not sure where to start? Take the ten-question placement test.