Perimeter as a Polynomial

A polynomial can describe a quantity that depends on an unknown. Suppose a rectangle has width $w$, and its length is 3 units longer, so the length is $w+3$. The perimeter is the distance around the rectangle: two lengths plus two widths.

The rectangle has two sides of length $w+3$ and two sides of length $w$. So its perimeter is $p = 2\left(w+3\right)+2w$. This formula works for every value of $w$.

Expand $2\left(w+3\right)+2w$ to get $2w+6+2w$. Combine the $w$ terms to get $p = 4w+6$. The two formulas give the same perimeter. The first one shows where each side comes from, and the second one is shorter to use.