Polynomial functions

$f(x) = 2x + 1$ is a straight line. $f(x) = x^2 - 3$ is a parabola. $f(x) = x^3 - 4x$ is a curve that changes direction twice. All three belong to the same family: nothing happens but adding, multiplying and raising to whole-number powers. This family is called polynomial functions, and because they are built so simply you can read a surprising amount off them before calculating anything.

The degree is the most important number

The degree is the highest exponent that appears. For $f(x) = 5x^3 - 2x + 7$ it is 3. It decides two things at once. First the number of roots: at most as many as the degree. So a function of degree 3 has at most three roots, but perhaps only one. Second the number of bends: a function of degree $n$ has at most $n - 1$ maxima and minima.

What happens far out

Substitute a very large number, say $x = 1000$. For $f(x) = x^3 - 4x + 7$ then $x^3$ is a billion, while $-4x$ contributes only $-4000$. The highest term outvotes all the others as soon as $x$ gets big enough. That is why the degree and the sign in front of it alone decide where the curve runs at the edges.

From this follows a rule with four cases. If the degree is even, both ends run the same way, as with a parabola: upwards for a positive leading coefficient, downwards for a negative one. If the degree is odd, the ends run apart: down on the left and up on the right, or the other way round. This also explains why every function of odd degree must have at least one root, because it comes from below and goes upwards and has to cross the axis on the way.

Finding roots

For roots there are three tools, and you try them in this order. First factoring out: in $x^3 - 4x = x(x^2 - 4)$ you see the root 0 at once, and then $\pm 2$. Second the quadratic formula, as soon as only degree 2 is left. Third guessing a root and lowering the degree by polynomial division, when the first two routes are not enough.

One special case is worth knowing: when only even exponents appear, as in $x^4 - 5x^2 + 4$, you set $u = x^2$ and get $u^2 - 5u + 4$. That is an ordinary quadratic equation with the solutions $u = 1$ and $u = 4$. Substituting back gives $x = \pm 1$ and $x = \pm 2$. This method is called substitution.