Polynomial and Rational Functions (HL)

Higher Level extends functions to polynomials of any degree and to the algebra of dividing them. A polynomial of degree $n$ has the form $P(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$. Its end behavior is governed by the leading term $a_n x^n$, and it has at most $n$ real roots and at most $n-1$ turning points.

The remainder and factor theorems

The remainder theorem states that when a polynomial $P(x)$ is divided by $(x - a)$, the remainder is simply $P(a)$. This lets you find a remainder without doing the full division — just evaluate.

The factor theorem is the special case where the remainder is zero: $(x - a)$ is a factor of $P(x)$ if and only if $P(a) = 0$. In other words, $a$ is a root exactly when $(x-a)$ divides $P(x)$ evenly. This is the primary tool for factoring cubics