2-1 Part I: Polynomials
Example polynomial:
3x2 - 2x + 1
3 terms:
quadratic
linear
constant
3x2
-2 x
+1
leading
coeff.
coefficient of
quadratic term
degree of
polynomial
coefficient of
linear term
leading term: term of highest degree; here it is 3x2
degree: degree of leading term
The tails of the graph of a polynomial:
2-1 Part I
p. 1
Polynomial Tail Principle:
for a polynomial, the tails will always go to + or - ,
AND tail behavior will be dictated by the leading term, as
follows:
leading term:
even
odd
leading coefficient
+
example: 3x4
example: -3x4
left tail  +
left tail  -
right tail  +
right tail  -
picture:
picture:
example: 3x3
left tail  -
right tail  +
picture:
example: -3x3
left tail  +
right tail  -
picture:
Summary:
even leading term: tails go same direction ( for +,  for -)
odd leading term: tails go opposite ( for +,  for -)
Example:
f(x) =
-7x2 - 3x4 + 7
leading term:
-3x4
even: go same direction coeff is -: direction is down ()
2-1 Part I
p. 2
Some example polynomials:
f(x) = x2 + 4x + 4
f(x) = 2x + 3

y






   






x


   




f(x) = -x3 + 4x - 1


x

f(x) = -x4 + x3 + 3x2 - x - 2
y


y








   





y





x

   





x








f(x) = .01x5 + .03x4 - .63x3 - .67x2 + 8.46x - 7.2

y



   




x
 




2-1 Part I
p. 3
Relationship of degree to x-intercepts, turning points
For a polynomial:
 degree > #turning points
 degree  #x-intercepts
The above graph has:
 5 turning points  at least degree 6
 6 zeros  at least degree 6
So, if it is the graph of a polynomial, it must be the graph
of a polynomial of degree 6 or greater
2-1 Part I
p. 4
Analyzing the graph or the formula of a polynomial
For a polynomial, we can deduce

things about the formula from the graph

things about the graph from the formula

y



   




x
 




tails:  polynomial is of odd degree, leading coeff +
degree > #turning points  degree > 4
degree  #x-intercepts  degree  5
remember the formula for this graph from the first page?
f(x) = .01x5 + .03x4 - .63x3 - .67x2 + 8.46x - 7.2
we can determine from the formula alone:
tails: , 5 > # turning points, 5  # x-intercepts
Another one: f(x) = -x4 + x3 + 3x2 - x - 2 (see page 1)
tails: 
degree > #turning points  4 > #turning points
degree  #x-intercepts  4  #x-intercepts
2-1 Part I
p. 5