Download CHAPTER 3: POLYNOMIAL AND RATIONAL FUNCTIONS

CHAPTER 3: POLYNOMIAL AND RATIONAL FUNCTIONS
3.1
POLYNOMIAL FUNCTIONS AND MODELING

Polynomial Function
A polynomial function P is given by
P( x)  an x n  an 1 x n 1  an  2 x n  2 
 a1 x  a0 ,
where the coefficients an , an1 ,..., a1 , a0 are real numbers and the
exponents are whole numbers.
o The first nonzero coefficient, an , is called the leading
coefficient
o The term an x n is called the leading term

o The degree of the polynomial function is n
The Leading Term Test
o The behavior of the graph of a polynomial function as x
becomes very large ( x   ) or very small ( x   ) is referred
to as the end behavior of the graph.
 The leading term determines a graph’s end behavior
o The Leading Term Test
If an x n is the leading term of a polynomial function, then the
behavior of the graph as x   or x   can be described in
one of the four following ways.
1.
If n is even, and an  0 :
2.
If n is even, and an  0 :

3.
If n is odd, and an  0 :
4.
If n is odd, and an  0 :
Even and Odd Multiplicity
If  x  c  , k  1, is a factor of a polynomial function P ( x ) and  x  c 
k
k 1
is not a factor and
o k is odd, then the graph crosses the x-axis at  c,0  ;
o k is even, then the graph is tangent to the x-axis at  c,0 .

3.2
Polynomial Models
o Cubic Regression
o Quartic Regression
GRAPHING POLYNOMIAL FUNCTIONS

Graphing Polynomial Functions
If P ( x ) is a polynomial function of degree n, the graph of the function
has at most n real zeros, and therefore at most n x-intercepts; at
most n 1 turning points.
o Steps for Graphing Polynomial Functions
1.
2.
3.
Use the leading-term test to determine the end behavior.
Find the zeros of the function by solving f ( x)  0. Any
real zeros are the first coordinates of the x-intercepts.
Use the zeros (x-intercepts) to divide the x-axis into
intervals and choose a test point in each interval to
determine the sign of all function values in that interval.
4.
Find f (0). This gives the y-intercept of the function.
5.
If necessary, find additional function values to determine
the general shape of the graph and then draw the graph.
As a partial check, use the facts that the graph has at
most n x-intercepts and at most n 1 turning points.
Multiplicity of zeros can also be considered in order to
check where the graph crosses or is tangent to the xaxis. We can also check the graph with a graphing
calculator.
6.

The Intermediate Value Theorem
For any polynomial function P ( x ) with real coefficients, suppose that
for a  b , P (a ) and P (b ) are of opposite signs. Then the function has
a real zero between a and b.
3.3
POLYNOMIAL DIVISION; THE REMAINDER AND FACTOR
THEOREMS

This section teaches us concepts that help to find the exact zeros of
polynomial functions with degree three or higher.

Consider the function
h( x)  x3  2x2  5x  6   x  3 x  1 x  2

This gives us the following zeros:
 x  3  0  x  3
 x  1  0  x  1
 x  2  0  x  2

3, 1, 2
When you divide one polynomial by another, you obtain a quotient
and a remainder.
o If the remainder is zero then the divisor is a factor of the
dividend.
P( x)  d ( x)  Q( x)  R( x)
P ( x) : Dividend
d ( x) : Divisor
Q( x) : Quotient
R ( x) : Remainder

Synthetic Division
o Consider the following:
 4x
3
 3x 2  x  7    x  2  .
A.
4 x 2  5 x  11
x  2 4 x3  3x 2 
x 7
4 x3  8 x 2
5x 2 
x
5x  10 x
2
11x  7
11x  22
29
B.
4 5 11
1 2 4  3  1 7
4 8
5 1
5  10
11  7
11  22
29
C.
2
4
3
1
7
8
10
22
5
11
29
4

The Remainder Theorem
If a number c is substituted for x in the polynomial f ( x) , then the
result f (c ) is the remainder that would be obtained by dividing
f ( x) by x  c. In other words, if f ( x)  ( x  c)  Q( x)  R, then
f (c)  R.

The Factor Theorem
For a polynomial f ( x) , if f (c)  0, then x  c is a factor of f ( x ).
Proof: If we divide f ( x) by x  c, we obtain a quotient and a
remainder, related as follows:
f ( x)  ( x  c)  Q( x)  f (c).
Then if f (c)  0, we have
f ( x)  ( x  c)  Q( x),
so x  c is a factor of f ( x) .
3.4
THEOREMS ABOUT ZEROS OF POLYNOMIAL FUNCTIONS

The Fundamental Theorem of Algebra
Every polynomial function of degree n, with n  1, has at least one
zero in the system of complex numbers.

Every polynomial function f of degree n, with n  1, can be factored
into n linear factors (not necessarily unique); that is,
f ( x)  an  x  c1  x  c2   x  cn  .

Finding Polynomials with Given Zeros
o If a complex number a  bi, b  0, is a zero of a polynomial
function f ( x) with real coefficients, then its conjugate,
a  bi , is also a zero.
 Example: Find a polynomial function of degree 3,
having the zeros 1, 3i, and 3i.
Solution: f ( x)  an  x 1 x  3i  x  3i  .
The number an can be any nonzero number.
The simplest function will be obtained if we
let an  1. Then we have
f ( x)   x  1 x  3i  x  3i 
  x  1  x 2  9 
 x3  9 x  x 2  9
 x3  x 2  9 x  9

Rational Coefficients
If a  b c , where a and b are rational and c is not a perfect square, is
a zero of a polynomial function f ( x) with rational coefficients, then
a  b c is also a zero.

Integer Coefficients and the Rational Zeros Theorem
o The Rational Zeros Theorem
Let P( x)  an x n  an 1 x n 1  an  2 x n  2   a1 x  a0 , where all the
coefficients are integers. Consider a rational number denoted
by p , where p and q are relatively prime (having no common
q
factor besides 1 and -1). If p is a zero of P ( x ) , then p is a
q
factor of a0 and q is a factor of an .

Example: Given f ( x)  3x 4  11x3  10 x  4 :
a) Find the rational zeros and then the other
zeros; that is, solve f ( x)  0.
b) Factor f ( x) into linear factors.
Solution:
a) Because the degree of f ( x) is 4, there are
at most 4 distinct zeros. The possibilities
for p are
q
possibilities for p
1, 2, 4
:
;
possibilities for q
1, 3
1
1 2
2 4
4
possibilities for p : 1,  1, 2,  2, 4,  4, ,  , ,  , , 
q
3
3 3
3 3
3
Now we need to graph the function on a
graphing calculator to see which of these
possibilities seem to be zeros.
When you divide the function by
x  (1) or x  1 , you’ll find that -1 is a zero.
Dividing the quotient 3x3  14 x 2  14 x  4
(obtained from the first division problem
1
1
above) by x 
will show that
is not a
3
3
2
2
zero. Repeating again for
shows that
3
3
is a zero. We obtain a quotient of
2
3 x 2  12 x  6 . So we have 1 and
as the
3
rational zeros and  3 x 2  12 x  6  factors as
3  x 2  4 x  2  , which yields 2  2 as the
b)
other two zeros.
Therefore the complete factorization of
f ( x) is
2

f ( x)   x  1  x    3 x 2  12 x  6 
3

2

  x  1  x    3  x 2  4 x  2 
3





2

 3  x  1  x    x  2  2   x  2  2 


3

3.5
RATIONAL FUNCTIONS

A rational function is a function f that is a quotient of two
p( x)
polynomials, that is, f ( x) 
, where p  x  and q  x  are polynomials
q( x)
and where q  x  is not the zero polynomial. The domain of f consists
of all inputs x for which q ( x)  0.

Asymptotes
o Vertical Asymptotes
 The line x  a is a vertical asymptote for the graph of f
if any of the following are true:
f ( x)   as x  a  or f ( x)   as x  a  , or
f ( x)   as x  a  or f ( x)   as x  a  .

Determining Vertical Asymptotes
p( x)
For a rational function f ( x) 
, where p ( x ) and
q( x)
q( x) are polynomials with no common factors other
than constants, if a is a zero of the denominator, then
the line x  a is a vertical asymptote for the graph of
the function.

Horizontal Asymptotes
 Determining a Horizontal Asymptote
 When the numerator and the denominator of
a rational function have the same degree,
a
the line y  is the horizontal asymptote,
b
where a and b are the leading coefficients
of the numerator and the denominator,
respectively.
 When the degree of the numerator of a
rational function is less than the degree of
the denominator, the x-axis, or y  0, is the

horizontal asymptote.
 When the degree of the numerator of a
rational function is greater than the degree
of the denominator, there is no horizontal
asymptote.
Oblique, or Slant, Asymptotes

Example: Find all asymptotes of
f ( x) 
2 x 2  3x  1
x2
Solution:
1. Since x  2  0  x  2 , this gives us a vertical
asymptote at the line x  2.
2. There are no horizontal asymptotes since
the degree of the numerator is greater than
the degree of the denominator.
3. Dividing: (2 x 2  3x  1)  ( x  2) , we get
1
. Now we see that when
 2 x  1 
x2
1
x   or x  ,
 0 and the value of
x2
f ( x)  2 x  1. This means that as the
absolute value of x becomes very large, the
graph of f ( x) gets very close to the graph
of y  2 x  1. Thus the line y  2 x  1 is the
3.6
oblique asymptote.
POLYNOMIAL AND RATIONAL INEQUALITIES

Polynomial Inequalities
o To solve a polynomial inequality:
1. Find an equivalent inequality with 0 on one side.
2. Solve the related polynomial equation.
3. Use the solutions to divide the x-axis into intervals.
Then select a test value from each interval and
determine the polynomial’s sign on the interval.
4. Determine the intervals for which the inequality is
satisfied and write interval notation or set-builder
notation for the solution set. Include the endpoints of
the intervals in the solution set if the inequality symbol is
 or  .

Rational Inequalities
o To solve a rational inequality:
1. Find an equivalent inequality with 0 on one side.
2. Change the inequality symbol to an equals sign and solve
the related equation.
3. Find the values of the variable for which the related
rational function is not defined.
4. The numbers found in steps (2) and (3) are called critical
values. Use the critical values to divide the x-axis into
intervals. Then test an x-value from each interval to
determine the function’s sign in that interval.
5. Select the intervals for which the inequality is satisfied
and write interval notation or set-builder notation for
the solution set. If the inequality symbol is  or , then
the solutions from step (2) should be included in the
solution set. The x-values found in step (3) are never
included in the solution set.
3.7
VARIATION AND APPLICATIONS

Direct Variation
If a situation gives rise to a linear function f ( x)  kx or y  kx, where k
is a positive constant, we say that we have direct variation, or that y
varies directly as x, or that y is directly proportional to x. The
number k is called the variation constant, or constant of
proportionality.

Inverse Variation
k
k
or y  , where k
x
x
is a positive constant, we say that we have inverse variation, or that y
varies inversely as x, or that y is inversely proportional to x. The
number k is called the variation constant, or constant of
proportionality.
If a situation gives rise to a linear function f ( x) 

Combined Variation
y varies directly as the nth power of x if there is some positive
constant k such that y  kx n .
y varies inversely as the nth power of x if there is some positive
k
constant k such that y  n .
x
y varies jointly as x and z if there is some positive constant k such
that y  kxz.