Polynomial Expansion (FOIL)

(x + a)(x + b) = x^2 + (a + b)x + ab — the FOIL pattern.

The idea

To expand a product of two sums is to multiply it out until no brackets remain. The rule is that every term of the first factor multiplies every term of the second:

$(a + b)(c + d) = ac + ad + bc + bd.$

There are four products, one for each pairing, and none may be skipped. The word FOIL names the pairs in order: First, Outer, Inner, Last.

The most common case is two factors of the form $(x + a)(x + b)$, with constants $a$ and $b$. The First product is $x \cdot x = x^{2}$; the Outer and Inner products, $bx$ and $ax$, are like terms and combine into a single middle term; the Last product is the constant $ab$. So

$(x + a)(x + b) = x^{2} + (a + b)x + ab.$

The middle coefficient is the sum of the two constants and the last term is their product, so the expansion can be written down directly.

Ways to work on it

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