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Mathematics 116
Chapter 4 Bittinger
• Polynomial
• and
• Rational Functions
Newt Gingrich
• “Perseverance is the hard
work you do after you get
tired of doing the hard work
you already did.”
Definition of a Polynomial
Function
• Polynomial function of x with degree n.
f ( x)  an x  an1 x
n
a x  a1 x  a0
2
2
1
n1


Joseph De Maistre (1753-1821 –
French Philosopher
• “It is one of man’s curious
idiosyncrasies to create
difficulties for the pleasure of
resolving them.”
Mathematics 116
• Polynomial Functions of
Higher Degree
Continuous
• The graph has no breaks, holes, or
gaps.
• Has only smooth rounded turns, not
sharp turns
• Its graph can be drawn with pencil
without lifting the pencil from the
paper.
Leading Coefficient Test
• The leading term determines the
“end behavior” of graphs.
• Very Important!
Objective
• Use the Leading Coefficient
Test to determine the end
behavior of graphs of
polynomial functions.
Intermediate Value Theorem
• Informal – Find a value x = a at which a
polynomial function is positive, and anther
value x = b at which it is negative, the
function has at least one real zero between
these two values.
• Use numerical zoom with table or
• Use [CAL] [1:zero]
Real Zeros of Polynomial
Functions
• x = a is a zero of function f
• x = a is a solution of the
polynomial equation f(x)=0
• (x-a) is a factor of the polynomial
f(x)
• (a,0) is an x-intercept of the graph
of f.
Repeated Zeros
• For a polynomial function, a factor
 x  a
k
,k 1
• Yields a repeated zero x = a of
multiplicity k
• If k is odd, the graph crosses at x = a
• If k is even, the graph touches at x=a
(not cross)
Objective
• Find and use zeros of
polynomial functions as
sketching aids.
Chinese Proverb:
• “A journey of a thousand
miles must begin with a
single step.”
Mathematics 116
• Real Zeros
• of
• Polynomial Functions
Objective
• Use long division to
divide polynomials by
other polynomials.
Objective
• Use synthetic division to
divide polynomials by
binomial of the form (x – k)
Reminder Theorem
• If a polynomial f(x) is divided
by x – k, the reminder is r = f(k)
Factor Theorem
• A polynomial f(x) has a factor
• (x – k) if and only if f(k) = 0
Using the remainder
• A reminder r obtained by dividing f(x)
by x – k
• 1. The reminder r gives the value of f
at
x=k
that is r = f(k)
• 2. If r = 0, (x – k) is a factor of f(x)
• 3. If r = 0, the (k,0) is an x intercept of
the
graph of f
• 4. If r = 0, then k is a root.
Rational Roots Test
• Possible rational zeros =
• factors of constant term
factors of leading coefficient
• Possible there are no rational
roots.
Descarte’s Rule of Signs
• Provides information on
number of positive roots and
number of negative roots.
William Cullen Bryant (1794-1878) U.S. poet,
editor
• “Difficulty, my brethren, is
the nurse of greatness – a
harsh nurse, who roughly
rocks her foster-children into
strength and athletic
proportion.”
Mathematics 116
• The
• Fundamental Theorem
• of
• Algebra
Number of roots
• A nth degree polynomial has n
roots.
• Some of these roots could be
multiple roots.
Linear Factorization Theorem
• Any nth-degree polynomial can
be written as the product of n
linear factors.
Objective
• Use the fundamental Theorem
of Algebra to determine the
number of zeros (roots) of a
polynomial function.
Objective
• Find all zeros of polynomial
functions including complex
zeros.
Conjugate Roots
• If a + bi, where b is not equal to
0
is a zero of a function f(x)
• the conjugate a – bi is also zero
of the function.
John F. Kennedy
• “We must use time as a
tool, not as a couch.”
Mathematics 116
• Rational Functions
• and
• Asymptotes
Rational Function
N ( x)
f ( x) 
D( x )
Graph – domain, range,
intercepts, asymptotes
1
f ( x) 
x
Graph – domain, range,
intercepts, asymptotes
1
f ( x)  2
x
Asymptotes
• Vertical
• Horizontal
• Slant
Objective
• Find the domains of rational
functions.
Objective
• Find horizontal and vertical
asymptotes of graphs of rational
functions.
Objective
• Use rational functions to model
and solve real-life problems.
George S. Patton
• “Accept the challenges,
so you may feel the
exhilaration of
victory.”
Mathematics 116
•Graphs of a Rational
Function
Graphing Rational Function
• 1. Simplify f if possible – reduce
• 2. Evaluate f(0) for y intercept and
plot
• 3. Find zeros or x intercepts – set
numerator = 0 & solve
• 4. Find vertical asymptotes – set
denominator = 0 and solve
• 5. Find horizontal / slant asymptotes
• 6. Find holes
Dan Rather
–“Courage is being afraid
but going on anyhow.”
College Algebra 116
•Quadratic
Inequalities
Sample Problem quadratic
inequalities #1
x  2x  8  0
2
 2,4
Sample Problem quadric
inequalities #2
6x  x  2
2
1


,


2

2 
,

 3 
Sample Problem quadratic
inequalities #3
x  6x  9  0
R   ,  
2
Sample Problem quadratic
inequalities #4
x4
0
x 1
(1,4]
Sample Problem quadratic
inequalities #5
3
2

0
x2 x3
 2,3  5, 
Everette Dennis – Media professor
• “There’s a compelling reason to
master information and news.
Clearly there will be better job
and financial opportunities.
Other high stakes will be
missed by people if they don’t
master and connect
information.”