TURNING POINTS OF POLYNOMIALS

Imagine yourself traveling along the graph of a polynomial, moving from left to right. Sometimes you go ‘uphill’, sometimes ‘downhill’, and sometimes you change direction.
Such a change of direction is called a turning point.

The purpose of this section is to make this concept more precise.
Roughly, a local maximum is a point on a curve where there is a ‘local high spot’.
That is, if you ‘stand on’ the point and don't look too far away,
then everything you see is lower (or at the same height).

If you look too far away (as the red arrow indicates, at right)
then you may see points that are higher.
People often say ‘local max’ instead of ‘local maximum’, for brevity.

(The precise definition of a local max is a bit complicated,
and is usually covered in a Calculus course.)

There is an analogous description for a local minimum (‘local min’, for short).
DEFINITION turning points of polynomials
A turning point of a polynomial is a point where there is a local max or a local min.
Notes about Turning Points:
A Horizontal Tangent Line
that is NOT a Turning Point

The following fact is easily proved in a Calculus course:

NUMBER OF POSSIBLE TURNING POINTS IN A POLYNOMIAL

A polynomial of degree $n$ can have at most $n-1$ turning points.

Notes:

Calculus results about derivatives, together with the Fundamental Theorem of Algebra,
will eventually firm up the concepts in this section.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
long division of polynomials
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11 12 13 14
AVAILABLE MASTERED IN PROGRESS

(MAX is 14; there are 14 different problem types.)