1
Pre-Calculus Notes
3.2, 3.6-3.7 – Polynomial Functions
Name _______________________
Date __________
A polynomial function is a function in the form f ( x)  an x n  an 1 x n 1  . . .  a1 x  a0 ,
Where an , an1 , . . . , a1 , a0 are real numbers and n is a non-negative integer.
The domain is the set of all real numbers.
The degree of the polynomial function is the largest power of x that appears in the function.
Example 1: Determine which of the following are polynomial functions.
2x 1
A.) f ( x)  4  2 x 3
B.) g ( x)  2
C.) h( x)  5
x 3
D.) F ( x)  0
F.) H ( x)  x( x  1) 2
E.) G ( x)  x  1
The graph of every polynomial function is a smooth, continuous curve. It will have no sharp corners or cusps,
and no gaps or holes.
Now graph an example of a non-smooth continuous curve
Graph an example of a smooth continuous curve
Include a place where there is a corner, a cusp and “jump
discontinuity” and a “removable discontinuity”
Label each part.
Identify each graph as a polynomial(P) or not a polynomial (N).
2
Power function: A power function of degree n is a function in the form f ( x)  ax , where a is a real number,
a  0 , and n is an integer, n > 0.
n
Label the graph with the power function that it is associated with. What happen to the graph as the power
increases?
Properties of power functions, y  x n , n is an even integer
 The graph is symmetric about the y-axis.
 The domain is the set of all real numbers. The range is the
set of nonnegative real numbers.
 The graph will always contain the points (0, 0), (1, 1), and
(1, 1).
 As the exponent n increases in magnitude, the graph
becomes more vertical when x <1 or x > 1, but for x nearer
the origin will flatten out and lie closer to the x-axis.
Properties of power functions, y  x n , n is an odd integer
 The graph is symmetric about the origin.
 The domain and range is the set of all real numbers.
 The graph will always contain the points (0, 0), (1, 1), and
(1, 1).
 As the exponent n increases in magnitude, the graph
becomes more vertical when x <1 or x > 1, but for x nearer
the origin will flatten out and lie closer to the x-axis.
Use transformations to graph the following functions.
Graph the parent power function and then graph using transformations.
1
( x  2) 4
2
A.) f ( x)  2  x 3
B.) f ( x) 
1
A.) f (x) = 2 + x 3
2
B.) f (x) = -2(x - 2)4
3
End behavior: The behavior of the graph of a function for very large or small values of x.
 f ( x)  ax n , for n odd will behave in the same way as f ( x)  ax 3 , with different end behaviors,
rising on one end and falling on the other.
 f ( x)  ax n , for n even will behave in the same fashion as f ( x)  ax 2 , with both ends rising or both
ends falling.
The following chart may be helpful:
Even and Positive
Even and Negative
Sketch the end behavior of the following graphs.
1. f ( x ) = 6x 3 +8x
Odd and Positive
2.
f ( x ) = 6x 3 -8x 4
End Behavior:
lim f ( x) =
End Behavior:
lim f ( x) =
lim f ( x) =
lim f ( x) =
x ®¥
Odd and Negative
x ®¥
x ® -¥
x ® -¥
End Behavior:
the left side goes __________
End Behavior:
the left side goes __________
the right side goes _________
the right side goes _________
3.
f ( x ) = -6x 5 +8x 2
End Behavior:
lim f ( x) =
x ®¥
lim
x ® -¥
f ( x) =
4.
f ( x ) = 8x 8 -8x 2
End Behavior:
lim f ( x) =
x ®¥
lim f ( x) =
x ® -¥
End Behavior:
the left side goes __________
End Behavior:
the left side goes __________
the right side goes _________
the right side goes _________
4
Recall: Even and Odd Functions and Symmetry
f is even if for every x in the domain, x is also in the domain and
Memorize!
f x  f  x
**graph is symmetric with respect to the y-axis
Even function:
f is odd if for every x in the domain, x is also in the domain and
Memorize!
 f x  f  x
**graph is symmetric with respect to the origin
Odd function:
EXAMPLES Determine algebraically whether a function is even, odd or neither.
Replace every x with –x and simplify;
If the result is equal to the original f(x), then the function is EVEN. This means f ( x)  f ( x)
If the result is the opposite of f(x), then the function is ODD. This means f ( x)   f ( x)
g  x   3x 4  2
First find g ( x)
f  x   4x3  2x
First find f ( x) then
a)
EXAMPLES
b)
find  f ( x)
Determine from a graph whether the function is even, odd or neither.
y
y
y












x












EVEN
Symmetric to x-axis
f ( x)  f ( x)
x
x
























ODD
Symmetric to origin
f ( x)   f ( x)
NEITHER
No symmetry
f ( x)  f ( x)
Now that we understand what happens when x gets really big (goes to infinity) and get really small (goes to
negative infinity). We will now concentrate on what happen in the middle of the graph. We are now going to
look at the real zeros of polynomial functions.
5
If f is a polynomial function and r is a real number for which f ( r )  0 , then r is called a real zero of f, or root
of f. If r is a real zero of f, then
 (r, 0) is an x-intercept of the graph of f.
 (x – r) is a factor of f.
Example: Find a polynomial function of degree 3, whose zeros are 2, 3, and 1.
If the same factor occurs more than once, then it is called a multiple factor of the function. The zero obtained
from the factor would be a multiple zero. The multiplicity of the zero is the number of times that it occurs.
Multiplicity of zeros can aid in graphing the function:
If r has multiplicity that is even, then it touches the graph at (r, 0), but does not cross the x-axis.
If r has multiplicity that is odd, then it crosses the x-axis at (r, 0).
Example: Determine the zeros and the multiplicity of each for the polynomial then graph
f ( x)  x( x  2) 2 ( x  3) 3
A polynomial function of degree n will have at most n -1 “turning points”, which occur at local maxima or local
minima.
f ( x)  x 2 ( x  3)( x  4)
x-int-___________ multiplicity-___________ cross or touches
x-int-___________ multiplicity-___________ cross or touches
x-int-___________ multiplicity-___________ cross or touches
y-int-_________ 
Symmetry-_________
End Behavior
Degree of polynomial _______
lim f (x) = _________
x ®¥
lim f (x) = _________
x ® -¥
Interval above x-axis___________________ Interval below x-axis________________
6
f ( x)  ( x  2) ( x  4)
3
2
x-int-___________ multiplicity-___________ cross or touches
x-int-___________ multiplicity-___________ cross or touches
x-int-___________ multiplicity-___________ cross or touches
y-int-_________ 
Symmetry-_____________
End Behavior
Degree of polynomial _______
lim f (x) = _________
x ®¥
lim f (x) = _________
x ® -¥
Interval above x-axis___________________ Interval below x-axis________________
g ( x)   x 2 ( x 2  9)( x  6)2
x-int-___________ multiplicity-___________ cross or touches
x-int-___________ multiplicity-___________ cross or touches
x-int-___________ multiplicity-___________ cross or touches
y-int-_________
Symmetry- _____________ 
End Behavior
Degree of polynomial _______
lim f (x) = _________
x ®¥
lim f (x) = _________
x ® -¥
Interval above x-axis___________________ Interval below x-axis________________
7
Summary:
The graph of a polynomial function, f ( x)  an x n  an 1 x n 1  . . .  a1 x  a0
 Degree: n
 Maximum number of turning points: n ─1
 At a zero of even multiplicity: graph touches x-axis
 At a zero of odd multiplicity: graph crosses x-axis
 Between zeros: graph either above or below x-axis
 End behavior: graph behaves (using even/odd rules) like the graphs of
f ( x)  ax 2 or f ( x)  ax 3
8
3.6 Real Zeros of a Polynomial
Division Algorithm:
f ( x)
r ( x)
or f ( x)  q( x)  g ( x)  r ( x)
 q( x) 
g ( x)
g ( x)
Dividend
quotient divisor
remainder
where r (x) is either the zero polynomial or a polynomial with degree less than that of g(x).
If the remainder is zero, then we say that f (x) divides evenly by g(x).
Example: Use synthetic division to find the remainder of: (3x3  2 x 2  4)  ( x  2) .
Example: Use long division to find the remainder of: (3x3  2 x 2  4)  ( x  2) .
9
Remainder Theorem: If f (x) is divided by (x – c), then the remainder is f (c).
Example: Use the Remainder Theorem and synthetic division to find the value of the function at the indicated values.
f ( x)  x 4  2 x 3  4 x 2  3 x  1
A.) f (3)
B.) f (─2)
Factor Theorem: Given a polynomial function f (x), x ─ c is a factor of f (x) if and only if
f (c) = 0.
This theorem has two parts:
 If f (c) = 0, then x ─ c is a factor of f (x).
 If x ─ c is a factor of f (x), then f (c) = 0.
Example: Use the Factor Theorem to determine whether the function,
f ( x)  4 x3  5 x 2  1 has the given factor
A.) x +2
B. x ─ 1
Theorem: A polynomial function of degree n will have no more than n zeros.
Descartes’ Rule of Signs:
Let f denote a polynomial function written in standard form.
 The number of positive real zeros of f either equals the number of variations in the sign of
the nonzero coefficients of f(x) or else equals that number less an even integer.
 The number of negative real zeros of f either equals the number of variations in the sign of
the nonzero coefficients of f(-x) or else equals that number less an even integer.
(Note: if the sum of the coefficients is zero, “1” is a zero of the function!)
Example: Use the number of real zeros theorem and Descartes’ Rule of Signs to discuss the real zeros of f(x).
f ( x)  2 x 6  x 4  2 x 3  3 x  1
10
Rational Zeros Theorem: If f (x) is a polynomial function with degree 1 or higher and integer
p
coefficients, then any rational zeros of f must be in the form , where p is a factor of the constant term and
q
q is a factor of the leading coefficient.
Example: List the possible rational zeros of f ( x)  2 x3  11x 2  7 x  6. Do not attempt to find the zeros
Upper Bound Theorem
If p(x) is divided by x – c and there are no sign changes in the quotient or remainder,
then c is upper bound.
Lower Bound Theorem
If p(x) is divided by x - c and there are alternating sign changes in the quotient and the remainder,
then c is the lower bound.
Example: Solve the equation in the real number system. f ( x)  x 3  8 x 2  11x  20
This means to do the following.
a) Use Descartes rule of sign to show how many possible positive and negative roots there are
b) Use Rational Zero Theorem to show what the possible rational zero’s are
c) Use synthetic or long division to test possible rational zero’s
d) Factor over the real number line. (x-a)(x-b)(x-c)=0
e) Solve over the real number line. (x = ??)
11
Example: Solve the equation in the real number system.
x  2 x  10 x  18 x  9  0
4
3
2
This means to do the following.
a) Use Descartes rule of sign to show how many possible positive and negative roots there are
b) Use Rational Zero Theorem to show what the possible rational zero’s are
c) Use synthetic or long division to test possible rational zero’s
d) Factor over the real number line. (x-a)(x-b)(x-c)=0
e) Solve over the real number line. (x = ??)
**Calculus**
The Intermediate Value Theorem: Let a and b be real numbers such that a < b. If f is a polynomial
function such that f(a) and f(b) are of opposite sign, then there is at least one zero of f between a and b.
Example: Use the I.V.T. to show that the polynomial function has a zero in the given interval.
f ( x)  3x 3  10 x  9; [3,  2]