Topics The anatomy of a polynomial Algebra and geometry of polynomials Polynomial Tail Principle Polynomial Turning Points and Intercepts Principle

Here's a typical polynomial:

Note: exponents in polynomials must be positive integers

Some terminology:

Polynomials can have one, two, three or more terms:

Polynomials can be of degree 0, 1, 2, 3, or more:

We refer to:

1. How a graph "behaves" at its extreme left and right is called tail behavior, and is a geometric (graphical) concept.

This graph has ·· turning points and ·· x-intercepts.

3. The degree of a polynomial and the sign of the leading coefficient are algebraic concepts:

Concerning the graph of a polynomial:

for a polynomial, the tails will always go to +oo or -oo

WHICH?

Just look at its leading term: (1) its degree - even or odd (2) its sign - plus or minus

leading term

Summary (polynomial tails)

Example: f(x) = -7x^{2} - 3x^{4} + 7

In general, a polynomial of lower degree cannot create a graph with lots of turning points; lots of turning points implies a polynomial of higher degree. The same kind of statement applies as well to x-intercepts: lots of x-intercepts imply higher degree.

It does not work the other way; a very high degree polynomial can produce a graph with few turning points and even no intercepts. The graph of f(x) = x^{20} looks pretty much like a parabola!

If a polynomial is of odd degree, it must have at least one x-intercept (it has to cross the x-axis, by the Polynomial Tail Principle).

Every polynomial f intercepts the y-axis somewhere (f(0) is always defined for a polynomial).

To be specific, we have the following principles for "going from algebra (degree) to geometry (turning points/intercepts":

Stating this same principle another way, for "going from geometry to algebra", we have:

(turning points/intercepts)-to-degree principle

Hint: remember "turning points is 1 less than degree" (roughly)

The graph of f can have at most ·· turning points, and ·· x-intercepts.

Application: going from the geometry of a polynomial to its algebra :

The above graph has: