2-1 Part I: Polynomial Functions

The anatomy of a polynomial

Here's a typical polynomial:

3x -  2x  +  1

Note: exponents in polynomials must be positive integers

Some terminology:

This polynomial has three terms.
The term having highest degree is called the leading term.
The degree of the polynomial is the degree of the leading term.

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:

3x2 + 2x + 1           as a quadratic trinomial
3x + 1                    as a linear binomial, etc.

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.

• for example, a graph may "shoot up" or "shoot down" at its extremes; i.e.,
• go off-scale to +oo or -oo
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:

f(x) = 7x2 - 3x4 + 7        (this is algebra)
has degree ·· and the sign of the leading coefficient is ··.

There are a couple of principles that tell us something about:
• the algebra of a polynomial, by examining its graph
• the graph of a polynomial, by examining its algebra

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

leading term: ··
leading term is even or odd? ··
tails go same or opposite ·· direction?
coefficient of leading term is positive or negative? ··
direction is up or down? ··

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)

• think of a parabola (the graph of a polynomial of degree 2)
• but it always has only 1 turning point!
Note:
• a polynomial of degree 6 could have 5,4,3,2, or 1 turning points
• a polynomial of degree 9 could have 9, 8, 7, ... x-intercepts
• but it must have at least 1 x-intercept . . .
• think about the polynomial tail principle!
• a polynomial with 3 turning points can only be of degree 4, 5, 6, ...
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:

·· turning points
·· x-intercepts
So, if it is the graph of a polynomial, it must be the graph of a polynomial of degree:
·· or greater