Functions - Properties of Polynomial Functions

A polynomial function of degree $n$ is defined by the expression $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$. Visually, these functions are always smooth and continuous curves with no breaks, holes, or sharp corners (cusps), extending infinitely in both directions along the x-axis.

The end behavior of a polynomial is determined by its degree $n$ and leading coefficient $a_n$. If $n$ is even, both ends of the graph point in the same vertical direction (forming a general 'U' or 'W' shape). If $n$ is odd, the ends point in opposite directions (forming a general 'S' or 'N' shape). Positive leading coefficients result in the right side of the graph pointing upwards.

The $y$-intercept of the polynomial is the point $(0, a_0)$, where the graph crosses the vertical axis. This is found by evaluating $P(0)$.

Real zeros or roots are the $x$-values where $P(x) = 0$, appearing as $x$-intercepts on the graph. The multiplicity of a root $k$ affects the graph's behavior: if $k$ is odd, the graph crosses the $x$-axis; if $k$ is even, the graph is tangent to the $x$-axis (touches and turns around), creating a local maximum or minimum at that point.

A polynomial of degree $n$ can have at most $n$ real roots and at most $n-1$ turning points. Turning points are the 'peaks' (local maxima) and 'valleys' (local minima) where the function changes from increasing to decreasing or vice-versa.

The Factor Theorem states that $(x - c)$ is a factor of $P(x)$ if and only if $P(c) = 0$. Geometrically, this means the graph passes exactly through the point $(c, 0)$. The Remainder Theorem states that the remainder of $\frac{P(x)}{x-c}$ is equal to the value $P(c)$.

Vieta's Formulas relate the coefficients of a polynomial to sums and products of its roots. For a polynomial of degree $n$, the sum of the roots is $-\frac{a_{n-1}}{a_n}$ and the product of the roots is $(-1)^n \frac{a_0}{a_n}$.