Hartfield – College Algebra
Unit FIVE | Page 1
(Version 2014-3D)
Topic 5-1a: Polynomial Expressions and Functions
More Terminology Recall
Recall the definitions of polynomials and terms.
Definition:
The degree of a term is equal to the
sum of the exponents of variables in
the term.
Definition:
The degree of a polynomial is equal
to the greatest degree for any term
in the polynomial.
Definition:
The leading term of a polynomial is
the term with the highest degree in a
one variable polynomial.
Definition:
The leading coefficient of a
polynomial is the coefficient of the
leading term.
Definition:
A constant term is a term whose
degree is zero and is usually
represented by a number.
Definition:
Descending order is an ordering
scheme for polynomials of one
variable where the terms are
arranged by degree from highest to
lowest.
Definition:
Definition:
A polynomial is a sum of terms.
A polynomial term is a product of
constants and variables with
non-negative integer exponents.
Which of the following represent polynomial
expressions? Of the expressions that are not
polynomials, identify the parts that are not properly
polynomial terms.
3
2
2x  6 x  x  5
x 4  3x 2  2
4
 6x  9
2
x
x100  100
5 x 3  7 x  0.5
x3 x2

x
3
2
Hartfield – College Algebra
Polynomial Functions and their parts
Definition:
Ex.
A polynomial function is a function
defined by a polynomial expression.
Answer the following questions about
the polynomial function.
f  x   3 x 2  6  2x  5 x 4
Unit FIVE | Page 2
(Version 2014-3D)
What is the simplest 8th degree polynomial
function?
How can you generalize an 8th degree monomial
function?
a. What is the degree of f?
b. What is the constant term of f?
th
True or False: Every 8 degree function has a term
of degree 5.
c. What is the leading term of f?
d. What is the leading coefficient of f?
e. Write f in descending order.
th
True or False: 8 degree functions can have a term
of degree 9.
Hartfield – College Algebra
Unit FIVE | Page 3
(Version 2014-3D)
General Polynomial Function Form
Turning Points
How can you generalize an 8th degree polynomial
function?
Definition:
A turning point of a graph is a point
where the function changes from
increasing to decreasing or vice
versa.
Turning points correspond
with local extrema (plural of
extremum).
How can you generalize an nth degree polynomial
function?
Maximum (sing.)
Maxima (pl.)
Minimum (sing.)
Minima (pl.)
An nth degree polynomial can have at most
n − 1 turning points. Polynomial functions with
k turning points must be at least degree (k + 1).
Functions of odd degree have an even number of turning
point and functions of even degree have an odd number
of turning points.
Hartfield – College Algebra
Topic 5-1: End Behavior of a Polynomial Function
End behavior is the term used to describe what
happens to function values as x gets very large
positive (x → ∞) or very large negative (x → −∞).
With polynomial functions, end behavior is always
either increasing without bound (graphically, going
up) or decreasing without bound (graphically, going
down).
(Version 2014-3D)
Unit FIVE | Page 4
The end behavior of a polynomial function is
determined by the leading coefficient and the
degree of the function.
1. The leading coefficient affects right end
behavior (x → ∞).
a. If the leading coefficient is positive, then the right
end of the graph will go up.
b. If the leading coefficient is negative, then the right
end of the graph will go down.
2. The function’s degree affects left end
behavior(x → −∞).
a. If the degree of the function is even, then the left
end of the graph will having matching behavior to
the right end of the graph.
b. If the degree of the function is odd, then the left
end of the graph will have opposite behavior to
the right end of the graph.
Hartfield – College Algebra
Ex. 1
Determine the end behavior of f and
justify your answer.
f  x   4x5  7x 2  1
Unit FIVE | Page 5
(Version 2014-3D)
Ex. 2
Determine the end behavior of f and
justify your answer.
f  x   5x3  4x 6  x
Hartfield – College Algebra
Topic 5-1c: Zeros of a Function
Definition: A zero of a function f is an x-value such
that f(x) = 0.
Unit FIVE | Page 6
(Version 2014-3D)
Ex. 1
Find the intercepts of each polynomial
function and discuss the end behavior.
P  x    x  1 x  1 x  2 
Recall the Zero Product Property: If A·B = 0, then
either A = 0 or B = 0. This means that factors of a
function can be solved to find the zeros of the
function.
x-intercepts on the graph of a function correspond
to zeros of the function. Thus a function in
factored form easily gives up its x-intercepts.
Ex.
Intercepts:
Find the zeros of f  x   x 2  6 x  5.
End behavior:
Hartfield – College Algebra
Ex. 1
Find the intercepts of each polynomial
function and discuss the end behavior.
P  x   2 x 3  x 2  x
Unit FIVE | Page 7
(Version 2014-3D)
Find a polynomial of leading coefficient of 1 with the
zeros as given. Write in factored form.
Ex. 1
−2, 1, 4
Degree 3
Ex. 2
−5, −1, 0, 3
Degree 4
Intercepts:
End behavior:
Hartfield – College Algebra
Topic 5-1d: Multiplicity of a Zero
Definition:
Theorem:
The multiplicity of a zero is the
number of times a zero of a function
occurs in the fully factored form of
the polynomial.
Unit FIVE | Page 8
(Version 2014-3D)
Ex. 1
Find the intercepts of each polynomial
function, describing the behavior of the
graph at each x-intercept, and discuss
the end behavior.
P x 
The sum of the multiplicities of all
zeros should be equal to the degree
of the function.
Effects of multiplicity on the graph of a function
Intercepts:
1. At an x-intercept corresponding to a zero of odd
multiplicity the graph of the function will cross the
x-axis.
2. At an x-intercept corresponding to a zero of even
multiplicity the graph of the function will turn at the
x-axis.
3. The higher the multiplicity of a zero, the greater the
localized “flatness” of the graph at the corresponding
x-intercept.
End behavior:
1
4
 x  1  x  2 
3
Hartfield – College Algebra
Ex. 2
Find the intercepts of each polynomial
function, describing the behavior of the
graph at each x-intercept, and discuss
the end behavior.
3
P  x    400
x 3  x  3  x  4 
Unit FIVE | Page 9
(Version 2014-3D)
Find a polynomial with the properties given below.
Write in factored form.
Ex. 1
Degree 4
Leading coefficient of −2
Zero −1, multiplicity 1
Zero 2, multiplicity 2
Zero 4, multiplicity 1
Ex. 2
Degree 6
Leading coefficient of
Zero −2, multiplicity 3
Zero 0, multiplicity 1
Zero 3, multiplicity 2
2
Intercepts:
End behavior:
1
8
Hartfield – College Algebra
Topic 5-2a:
Long Division of Polynomials
Recall how to do long division of numbers.
7109
23
Unit FIVE | Page 10
(Version 2014-3D)
Long Division Algorithm for Polynomials
1. Divide the leading term of the divisor into the leading
term of the dividend. Write in the quotient.
2. Multiply the result of step 1 to the divisor and write
under the dividend.
3. Subtract.
4. Repeat steps 1-3 by bringing down terms of the
dividend after subtraction. The algorithm is
completed once the remaining leading term under
the dividend is of lower degree than the degree of
the divisor.
To check your division:
 quotient  divisor   remainder  numerator
Hartfield – College Algebra
Ex. 1
Divide. Identify the quotient and
remainder.
x 3  5 x 2  2x  3
x 2
(Version 2014-3D)
Unit FIVE | Page 11
Hartfield – College Algebra
Ex. 2
Divide. Identify the quotient and
remainder.
2x 4  6 x 3  2x  7
x 2  2x  2
(Version 2014-3D)
Unit FIVE | Page 12
Hartfield – College Algebra
Ex. 3
Divide. Identify the quotient and
remainder.
4x3  8x 2  6x  5
2x  1
(Version 2014-3D)
Unit FIVE | Page 13
Hartfield – College Algebra
Unit FIVE | Page 14
(Version 2014-3D)
Topic 5-2b: Synthetic Division of Polynomials
Synthetic Division Algorithm for Polynomials
In addition to the long division, there is another
method for dividing polynomials called synthetic
division.
For a division problem in the form
Drawbacks to Synthetic Division
Rarely feasible if the denominator is nonlinear.
Requires extra manipulation if the leading
coefficient of a linear denominator is not 1.
Advantages to Synthetic Division
Far easier, quicker, and less messy than long
division.
Most of the cases where division is useful can
be handled by synthetic division.
set up the problem as
c an an a
P( x )
,
x c
 a2 a1 a0
where an through a0 are the coefficients of P.
1. Bring the first remaining coefficient down under the
second line.
2. Multiply the number under the second line to c.
Write the product under the next coefficient of P.
3. Add and write the sum under the second line.
4. Repeat steps 2 and 3 until the coefficients of P are
exhausted.
5. Draw a box around the last number under the
second line. This is the remainder.
6. The preceding numbers under the second line are
the coefficients of the quotient which should be
exactly one degree less than P.
Hartfield – College Algebra
Ex. 1
Divide. Identify the quotient and
remainder.
x 3  5 x 2  2x  3
x 2
Unit FIVE | Page 15
(Version 2014-3D)
Ex. 2
Divide. Identify the quotient and
remainder.
2x 4  5 x 3  2x  9
x 3
Hartfield – College Algebra
Ex. 3
Divide. Identify the quotient and
remainder.
x 3  8 x 2  2x
x 1
Unit FIVE | Page 16
(Version 2014-3D)
Ex. 4
Divide. Identify the quotient and
remainder.
x 3  9 x 2  22 x  8
x4
Hartfield – College Algebra
Topic 5-2c:
Remainder and Factor Theorems
1. The remainder found by synthetic division is
equal to P(c).
(Remainder Theorem)
Use previous examples to evaluate:
P ( x )  x 3  5 x 2  2 x  3 at x = 2
P ( x )  2 x 4  5 x 3  2 x  9 at x = −3
P ( x )  x 3  8 x 2  2 x at x = -1
(Version 2014-3D)
Unit FIVE | Page 17
2. If the remainder found by division is zero, then
Q(x) is a factor of P(x). (In other words, Q
times something is P).
Hartfield – College Algebra
Topic 5-3a: Rational Functions, part 1
Definition:
A rational function is defined by the
quotient of two polynomial functions.
Unit FIVE | Page 18
(Version 2014-3D)
Intercepts and Vertical Asymptotes of Rational
Functions
The y-intercept occurs at R(0).
The x-intercept(s) occur wherever
P(x) = 0 but Q(x) ≠ 0.
Let R represent a rational function and P and Q
P x
represent polynomial functions, then R  x  
.
Qx
(That is, an x-value makes the numerator equal 0 but
not the denominator.)
Definition:
Vertical Asymptotes occur wherever
Q(x) = 0 but P(x) ≠ 0.
An asymptote is a line that a graph
approaches but does not intersect
as values approach ±∞.
(That is, an x-value makes the denominator equal 0 but
not the numerator.)
A vertical asymptote of a rational function is a line x = h
where h is a solution to Q(x) = 0 and not P(x) = 0.
Hartfield – College Algebra
Ex.
(Version 2014-3D)
Find the intercepts and vertical
asymptote(s) of R.
2
x  4 x  12
R x  2
x  3x  4
Unit FIVE | Page 19
Horizontal Asymptotes of Rational Functions
Horizontal Asymptotes correspond with end
behavior. Unlike polynomial functions which either
increase or decrease without bound as x  ,
rational functions may approach a fixed #.
To determine if a rational function has a horizontal
asymptote, identify the degrees of the numerator
and denominator, then:
a: If the degree of the numerator > the degree of the
denominator, then the rational function does not
have a horizontal asymptote.
b: If the degree of the numerator < the degree of the
denominator, then the rational function has a
horizontal asymptote of y = 0.
c: If the degree of the numerator = the degree of the
denominator, then the rational function has a
horizontal asymptote of y = k where k is the ratio of
the leading coefficients of P and Q.
Hartfield – College Algebra
Ex. 1
Unit FIVE | Page 20
(Version 2014-3D)
Identify the attributes of the rational
function and then sketch a graph.
R x 
6  2x
x 1
y-intercept:
x-intercept(s):
vertical asymptote(s):
y-int:
horizontal asymptote:
x-int:
VA:
HA:
Hartfield – College Algebra
Ex. 2
Unit FIVE | Page 21
(Version 2014-3D)
Identify the attributes of the rational
function and then sketch a graph.
R x 
x 2
x 2  3x  4
y-intercept:
x-intercept(s):
vertical asymptote(s):
y-int:
horizontal asymptote:
x-int:
VA:
HA:
Hartfield – College Algebra
Ex. 3
Unit FIVE | Page 22
(Version 2014-3D)
Identify the attributes of the rational
function and then sketch a graph.
2 x 2  18
R x  2
x  7x  6
y-intercept:
x-intercept(s):
vertical asymptote(s):
y-int:
horizontal asymptote:
x-int:
VA:
HA:
Hartfield – College Algebra
Topic 5-3b:
Rational Functions, part 2
(Version 2014-3D)
Unit FIVE | Page 23
Holes of a Rational Function
Definition: A slant asymptote, also called an
oblique asymptote, is an asymptote that
is neither horizontal nor vertical.
A hole in a function is created when the rule of the
rational function can be simplified and the
denominator causes a point-wise loss of definition.
Slant asymptotes occur whenever the numerator is
exactly one degree greater than the denominator.
Holes occur wherever P(x) = 0 and Q(x) = 0.
The quotient found by dividing the rational
expression defines the rule of the slope of the slant
asymptote (i.e. y = quotient).
With rational functions, it is not possible to have
both a horizontal asymptote and a slant asymptote.
(That is, an x-value makes both the numerator and
denominator equal 0.)
To find the ordered pair coordinates of a hole, set
the common factor of P and Q equal to zero and
solve. The solution will be the x-value of the hole.
To find the y-value of the hole, simplify the rational
expression and evaluate the x-value of the hole.
Hartfield – College Algebra
Ex. 4
Unit FIVE | Page 24
(Version 2014-3D)
Identify the attributes of the rational
function and then sketch a graph.
2x 2  4 x  6
R x 
x 2
y-intercept:
x-intercept(s):
vertical asymptote(s):
y-int:
horizontal/slant asymptote:
x-int:
VA:
HA/SA:
hole(s):
hole:
Hartfield – College Algebra
Ex. 5
Unit FIVE | Page 25
(Version 2014-3D)
Identify the attributes of the rational
function and then sketch a graph.
4  x2
R x  2
x  2x  8
y-intercept:
x-intercept(s):
vertical asymptote(s):
y-int:
horizontal/slant asymptote:
x-int:
VA:
HA/SA:
hole(s):
hole: