Polynomials

A polynomial is simply a few terms added together, like $4x^3 - x + 7$, where each term is a number (the coefficient) times $x$ raised to a whole-number power. The name means "many terms" ("poly" is Greek for many). Polynomials are the simplest formulas you can build from $x$ using only adding and multiplying, which is why they show up everywhere. All we're doing in this lesson is adding and multiplying them. The formula $(a + b)^2 = a^2 + 2ab + b^2$ looks like something to memorize, but it's just $a^2$ (a times a), $b^2$ (b times b) and $2ab$ (a times b, which happens twice). Nothing more is going on.

Vocabulary. The degree is the highest power, and the leading coefficient is the number in front of it. So $4x^3 - x + 7$ has degree 3 and leading coefficient 4. The degree matters more than it looks: for large $x$ the top term swamps everything else, so it decides how the whole thing behaves.

Adding and subtracting. Combine like terms, meaning terms with the same power. It's the same as adding $3$ apples to $2$ apples; $x^2$ and $x^3$ are different fruit. When you subtract a whole polynomial, the minus sign hits every term inside the parentheses, not just the first one.

Multiplying. This is the distributive law, used over and over: every term in the first factor multiplies every term in the second. For two binomials that's the FOIL pattern. A few products come up so often they're worth knowing on sight: $(a + b)^2$, $(a - b)^2$, and $(a + b)(a - b) = a^2 - b^2$, where the middle terms cancel.

A common mistake. It's tempting to square $(a + b)$ term by term and get $a^2 + b^2$. That drops the cross term $2ab$, which comes from $a$ meeting $b$ twice, once each way. The picture below shows those two missing pieces as actual rectangles.

Polynomials model projectile heights, profit curves, and the smooth curves (splines) used in fonts and animation.