2-1 Part I: Polynomial Functions

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


The anatomy of a polynomial

Here's a typical polynomial:

3x2  -  2x  +  1

Note: exponents in polynomials must be positive integers

Some terminology:

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

#terms name examples
one term monomial 3x2, 2x, 3
two terms binomial 2x + y, 3x2 - 1 
three terms trinomial  3x - 2y + 5

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

degree name examples
0 constant 3
1 linear 3x, 2y + 1
2 quadratic  2x2, -3x2 + 1 
3 cubic 2x3, -3x3 + 2x 

We refer to:

Algebra and geometry of polynomials

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

2.  The number of turning points and number of x-intercepts of a graph are geometric concepts.  Here's some geometry:

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

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

has degree ·· and the sign of the leading coefficient is ··.
There are a couple of principles that tell us something about:
The Polynomial Tail Principle

Concerning the graph of a polynomial:

left tail – behavior at extreme left of the graph
right tail – behavior at extreme right of graph

Polynomial Tail Principle

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

  sign + sign -
degree even example: 3x4
left tail => +oo
right tail => +oo
picture: 

example: -3x4
left tail => -oo
right tail => -oo
picture: 

degree odd example: 3x3
left tail => -oo
right tail => +¥
picture:
example: -3x3
left tail => +oo
right tail => -¥
picture: 

Summary (polynomial tails)


 even leading term: tails go same direction (up for +, down for -)
 odd leading term: tails go opposite (down-up for +, up-down for -)

Example:                 f(x) = -7x2 - 3x4 + 7

The Polynomial Turning Points and Intercepts Principle

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) = x20 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":
 

degree-to-(turning points/intercepts) principle

 A polynomial of degree n has:
    n – 1 turning points or fewer
    n x-intercepts or fewer
    exactly 1 y-intercept
 Furthermore, if n is odd, it must have:
    at least 1 x-intercept

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

(turning points/intercepts)-to-degree principle


 A polynomial having n turning points has:
    degree n + 1 or greater
 A polynomial having n intercepts has:
    degree n or greater

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

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

Example:            f(x) = -7x2 - 3x4 + 7

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:

So, if it is the graph of a polynomial, it must be the graph of a polynomial of degree:
·· or greater