2.3 Polynomial Functions of Higher Degrees & Modeling

A polynomial written in standard form is written with descending degrees.

When graphing transformations, look for the y-intercept (f(0)) and take into account the transformations.

Standard form of a cubic. y=a x3+b x2+cx+d.

Every polynomial is defined as continuous for all real numbers. Not only are graphs of polynomials unbroken without jumps or holes, but they are smooth, unbroken lines or curves, with no sharp corners or cusps.

In sufficiently large viewing windows, the graph of a polynomial and the graph of its leading term appear to be identical. This means the leading term dominates in determining the end behavior of the polynomial.

Theorem: A polynomial function of degree n has at most n−1 local extreme and at most n zeros. Some of the zeros could be single, double (aka repeated zero), triple roots, etc. Double roots are tangent to the x-axis, but don’t cross it. Triple roots cross the x-axis and flatten as they do so.

The end behavior of a polynomial is closely related to the end behavior of its leading term.

Theorem: If P(x) is a polynomial of odd degree with real coefficients, then the equation P(x) = 0 has at least one real root.

Odd-degreed polynomials have end behavior that go in opposite directions and even-degreed polynomials have end behavior that go in the same direction.