Transcript Slide 1

Polynomial and
Rational
Functions
Chapter 3
Quadratic
Functions and
Models
Section 3.1
Quadratic Functions

Quadratic function: Function of the
form
f(x) = ax2 + bx + c
(a, b and c real numbers, a ≠ 0)
Quadratic Functions

Example. Plot the graphs of f(x) =
x2, g(x) = 3x2 and
30
20
10
-10
-8
-6
-4
-2
2
-10
-20
-30
4
6
8
10
Quadratic Functions

Example. Plot the graphs of f(x) =
{x2, g(x) = {3x2 and
30
20
10
-10
-8
-6
-4
-2
2
-10
-20
-30
4
6
8
10
Parabolas

Parabola: The graph of a quadratic
function
If a > 0, the parabola opens up
 If a < 0, the parabola opens down
 Vertex: highest / lowest point of a
parabola

Parabolas

Axis of symmetry: Vertical line
passing through the vertex
Parabolas

Example. For the function
f(x) = {3x2 +12x { 11
(a) Problem: Graph the function
Answer:
10
8
6
4
2
-10
-8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
10
Parabolas

Example. (cont.)
(b) Problem: Find the vertex and axis of
symmetry.
Answer:
Parabolas

Locations of vertex and axis of
symmetry:

Set

Set

Vertex is at:

Axis of symmetry runs through vertex

Parabolas

Example. For the parabola defined by
f(x) = 2x2 { 3x + 2
(a) Problem: Without graphing, locate the
vertex.
Answer:
(b) Problem: Does the parabola open up
or down?
Answer:
x-intercepts of a Parabola

For a quadratic function
f(x) = ax2 + bx + c:

Discriminant is b2 { 4ac.

Number of x-intercepts depends on the
discriminant.

Positive discriminant: Two x-intercepts

Negative discriminant: Zero x-intercepts

Zero discriminant: One x-intercept
(Vertex lies on x-axis)
x-intercepts of a Parabola
Graphing Quadratic Functions

Example. For the function
f(x) = 2x2 + 8x + 4
(a) Problem: Find the vertex
Answer:
(b) Problem: Find the intercepts.
Answer:
Graphing Quadratic Functions

Example. (cont.)
(c) Problem: Graph the function
Answer:
10
8
6
4
2
-10
-8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
10
Graphing Quadratic Functions

Example. (cont.)
(d) Problem: Determine the domain and
range of f.
Answer:
(e) Problem: Determine where f is
increasing and decreasing.
Answer:
Graphing Quadratic Functions

Example.
Problem: Determine the quadratic
function whose vertex is (2, 3) and
whose y-intercept is 11.
Answer:
14
12
10
8
6
4
2
-14 -12 -10 -8 -6 -4 -2
-2
-4
-6
-8
-10
-12
-14
2
4
6
8
10 12 14
Graphing Quadratic Functions

Method 1 for Graphing


Complete the square in x to write the
quadratic function in the form
y = a(x { h)2 + k
Graph the function using
transformations
Graphing Quadratic Functions

Method 2 for Graphing




Determine the vertex
Determine the axis of symmetry
Determine the y-intercept f(0)
Find the discriminant b2 { 4ac.



If b2 { 4ac > 0, two x-intercepts
If b2 { 4ac = 0, one x-intercept
(at the vertex)
If b2 { 4ac < 0, no x-intercepts.
Graphing Quadratic Functions

Method 2 for Graphing

Find an additional point


Use the y-intercept and axis of symmetry.
Plot the points and draw the graph
Graphing Quadratic Functions

Example. For the quadratic function
f(x) = 3x2 { 12x + 7
(a) Problem: Determine whether f has a
maximum or minimum value, then find
it.
Answer:
Graphing Quadratic Functions

Example. (cont.)
(b) Problem: Graph f
Answer:
10
8
6
4
2
-10
-8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
10
Quadratic Relations
Quadratic Relations

Example. An engineer collects the
following data showing the speed s of
a Ford Taurus and its average miles
per gallon, M.
Quadratic Relations
Speed, s
Miles per Gallon, M
30
18
35
20
40
23
40
25
45
25
50
28
55
30
60
29
65
26
65
25
70
25
Quadratic Relations

Example. (cont.)
(a) Problem: Draw a scatter diagram of
the data
Answer:
Quadratic Relations

Example. (cont.)
(b) Problem: Find the quadratic function
of best fit to these data.
Answer:
Quadratic Relations

Example. (cont.)
(c) Problem: Use the function to
determine the speed that maximizes
miles per gallon.
Answer:
Key Points





Quadratic Functions
Parabolas
x-intercepts of a Parabola
Graphing Quadratic Functions
Quadratic Relations
Polynomial
Functions and
Models
Section 3.2
Polynomial Functions

Polynomial function: Function of the form
f(x) = anxn + an {1xn {1 +  + a1x + a0

an, an {1, …, a1, a0 real numbers

n is a nonnegative integer (an  0)

Domain is the set of all real numbers

Terminology

Leading coefficient: an

Degree: n (largest power)

Constant term: a0
Polynomial Functions

Degrees:




Zero function: undefined degree
Constant functions: degree 0.
(Non-constant) linear functions: degree 1.
Quadratic functions: degree 2.
Polynomial Functions

Example. Determine which of the
following are polynomial functions? For
those that are, find the degree.
(a) Problem: f(x) = 3x + 6x2
Answer:
(b) Problem: g(x) = 13x3 + 5 + 9x4
Answer:
(c) Problem: h(x) = 14
Answer:
(d) Problem:
Answer:
Polynomial Functions

Graph of a polynomial function will be
smooth and continuous.


Smooth: no sharp corners or cusps.
Continuous: no gaps or holes.
Power Functions

Power function of degree n:

Function of the form
f(x) = axn
a  0 a real number
 n > 0 is an integer.

Power Functions

The graph depends on whether n is
even or odd.
Power Functions

Properties of f(x) = axn

Symmetry:

If n is even, f is even.

If n is odd, f is odd.

Domain: All real numbers.

Range:

If n is even, All nonnegative real numbers

If n is odd, All real numbers.
Power Functions

Properties of f(x) = axn


Points on graph:

If n is even: (0, 0), (1, 1) and ({1, 1)

If n is odd: (0, 0), (1, 1) and ({1, {1)
Shape: As n increases

Graph becomes more vertical if |x| > 1

More horizontal near origin
Graphing Using Transformations

Example.
Problem: Graph f(x) = (x { 1)4
Answer:
4
2
-4
-2
2
-2
-4
4
Graphing Using Transformations

Example.
Problem: Graph f(x) = x5 + 2
Answer:
4
2
-4
-2
2
-2
-4
4
Zeros of a Polynomial

Zero or root of a polynomial f:
r a real number for which f(r) = 0
 r is an x-intercept of the graph of f.
 (x { r) is a factor of f.

Zeros of a Polynomial
Zeros of a Polynomial

Example.
Problem: Find a polynomial of degree 3
whose zeros are {4, {2 and 3.
Answer:
40
30
20
10
-10
-5
5
-10
-20
-30
-40
10
Zeros of a Polynomial

Repeated or multiple zero or root of
f:
Same factor (x { r) appears more than
once
 Zero of multiplicity m:


(x { r)m is a factor of f and (x { r)m+1 isn’t.
Zeros of a Polynomial

Example.
Problem: For the polynomial, list all zeros
and their multiplicities.
f(x) = {2(x { 2)(x + 1)3(x { 3)4
Answer:
Zeros of a Polynomial

Example. For the polynomial
f(x) = {x3(x { 3)2(x + 2)
(a) Problem: Graph the polynomial
Answer:
40
20
-4
-2
2
-20
-40
4
Zeros of a Polynomial

Example. (cont.)
(b) Problem: Find the zeros and their
multiplicities
Answer:
Multiplicity

Role of multiplicity:

r a zero of even multiplicity:
f(x) does not change sign at r
 Graph touches the x-axis at r, but does not
cross

40
20
-4
-2
2
-20
-40
4
Multiplicity

Role of multiplicity:

r a zero of odd multiplicity:
f(x) changes sign at r
 Graph crosses x-axis at r

40
20
-4
-2
2
-20
-40
4
Turning Points

Turning points:
Points where graph changes from
increasing to decreasing function or vice
versa
 Turning points correspond to local
extrema.


Theorem.
If f is a polynomial function of degree
n, then f has at most n { 1 turning
points.
End Behavior
Theorem. [End Behavior]
For large values of x, either positive
or negative, that is, for large |x|, the
graph of the polynomial
f(x) = anxn + an{1xn{1 +  + a1x + a0
resembles the graph of the power
function
y = a n xn

End Behavior
End behavior of:
f(x) = anxn + an{1xn{1 +  + a1x + a0

Analyzing Polynomial Graphs

Example. For the polynomial:
f(x) =12x3 { 2x4 { 2x5
(a) Problem: Find the degree.
Answer:
(b) Problem: Determine the end behavior.
(Find the power function that the graph
of f resembles for large values of |x|.)
Answer:
Analyzing Polynomial Graphs

Example. (cont.)
(c) Problem: Find the x-intercept(s), if
any
Answer:
(d) Problem: Find the y-intercept.
Answer:
(e) Problem: Does the graph cross or
touch the x-axis at each x-intercept:
Answer:
Analyzing Polynomial Graphs

Example. (cont.)
(f) Problem: Graph f using a graphing
utility
80
Answer:
60
40
20
-4
-2
2
-20
-40
-60
-80
4
Analyzing Polynomial Graphs

Example. (cont.)
(g) Problem: Determine the number of
turning points on the graph of f.
Approximate the turning points to 2
decimal places.
Answer:
(h) Problem: Find the domain
Answer:
Analyzing Polynomial Graphs

Example. (cont.)
(i) Problem: Find the range
Answer:
(j) Problem: Find where f is increasing
Answer:
(k) Problem: Find where f is decreasing
Answer:
Cubic Relations
Cubic Relations

Example. The following data
represent the average number of miles
driven (in thousands) annually by
vans, pickups, and sports utility
vehicles for the years 1993-2001,
where x = 1 represents 1993, x = 2
represents 1994, and so on.
Cubic Relations
Year, x
Average Miles Driven, M
1993, 1
12.4
1994, 2
12.2
1995, 3
12.0
1996, 4
11.8
1997, 5
12.1
1998, 6
12.2
1999, 7
12.0
2000, 8
11.7
2001, 9
11.1
Cubic Relations

Example. (cont.)
(a) Problem: Draw a scatter diagram of
the data using x as the independent
variable and M as the dependent
variable.
Answer:
Cubic Relations

Example. (cont.)
(b) Problem: Find the cubic function of
best fit and graph it
Answer:
Key Points









Polynomial Functions
Power Functions
Graphing Using Transformations
Zeros of a Polynomial
Multiplicity
Turning Points
End Behavior
Analyzing Polynomial Graphs
Cubic Relations
The Real Zeros of
a Polynomial
Function
Section 3.6
Division Algorithm

Theorem. [Division Algorithm]
If f(x) and g(x) denote polynomial
functions and if g(x) is a polynomial whose
degree is greater than zero, then there are
unique polynomial functions q(x) and r(x)
such that
where r(x) is either the zero polynomial or
a polynomial of degree less than that of
g(x).
Division Algorithm

Division algorithm
f(x) is the dividend
 q(x) is the quotient
 g(x) is the divisor
 r(x) is the remainder

Remainder Theorem

First-degree divisor
Has form g(x) = x { c
 Remainder r(x)

Either the zero polynomial or a polynomial
of degree 0,
 Either way a number R.

Becomes f(x) = (x { c)q(x) + R
 Substitute x = c
 Becomes f(c) = R

Remainder Theorem

Theorem. [Remainder Theorem]
Let f be a polynomial function. If
f(x) is divided by x { c, the
remainder is f(c).
Remainder Theorem

Example. Find the remainder if
f(x) = x3 + 3x2 + 2x { 6
is divided by:
(a) Problem: x + 2
Answer:
(b) Problem: x { 1
Answer:
Factor Theorem



Theorem. [Factor Theorem]
Let f be a polynomial function. Then
x { c is a factor of f(x) if and only if
f(c) = 0.
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.
Factor Theorem

Example. Determine whether the
function
f(x) = {2x3 { x2 + 4x + 3
has the given factor:
(a) Problem: x + 1
Answer:
(b) Problem: x { 1
Answer:
Number of Real Zeros

Theorem. [Number of Real Zeros]
A polynomial function of degree n,
n ¸ 1, has at most n real zeros.
Rational Zeros Theorem
Theorem. [Rational Zeros Theorem]
Let f be a polynomial function of
degree 1 or higher of the form
f(x) = anxn + an{1xn{1 +  + a1x + a0
an  0, a0  0, where each coefficient
is an integer. If p/q, in lowest terms,
is a rational zero of f, then p must be
a factor of a0 and q must be a factor
of an.

Rational Zeros Theorem

Example.
Problem: List the potential rational zeros
of
f(x) = 3x3 + 8x2 { 7x { 12
Answer:
Finding Zeros of a Polynomial

Determine the maximum number of
zeros.


If the polynomial has integer
coefficients:


Degree of the polynomial
Use the Rational Zeros Theorem to find
potential rational zeros
Using a graphing utility, graph the
function.
Finding Zeros of a Polynomial

Test values
Test a potential rational zero
 Each time a zero is found, repeat on the
depressed equation.

Finding Zeros of a Polynomial

Example.
Problem: Find the rational zeros of the
polynomial in the last example.
f(x) = 3x3 + 8x2 { 7x { 12
Answer:
Finding Zeros of a Polynomial

Example.
Problem: Find the real zeros of
f(x) = 2x4 + 13x3 + 29x2 + 27x + 9
and write f in factored form
Answer:
Factoring Polynomials



Irreducible quadratic: Cannot be factored
over the real numbers
Theorem.
Every polynomial function (with real
coefficients) can be uniquely factored into a
product of linear factors and irreducible
quadratic factors
Corollary.
A polynomial function (with real
coefficients) of odd degree has at least one
real zero
Factoring Polynomials

Example.
Problem: Factor
f(x)=2x5 { 9x4 + 20x3 { 40x2 + 48x {16
Answer:
Bounds on Zeros

Bound on the zeros of a polynomial
Positive number M
 Every zero lies between {M and M.

Bounds on Zeros
Theorem. [Bounds on Zeros]
Let f denote a polynomial whose
leading coefficient is 1.
f(x) = xn + an{1xn{1 +  + a1x + a0
A bound M on the zeros of f is the
smaller of the two numbers
Max{1, ja0j + ja1j +  + jan-1j},
1 + Max{ja0j ,ja1j , … , jan-1j}

Bounds on Zeros

Example. Find a bound to the zeros
of each polynomial.
(a) Problem:
f(x) = x5 + 6x3 { 7x2 + 8x { 10
Answer:
(b) Problem:
g(x) = 3x5 { 4x4 + 2x3 + x2 +5
Answer:
Intermediate Value Theorem

Theorem. [Intermediate Value Theorem]
Let f denote a continuous function. If
a < b and if f(a) and f(b) are of
opposite sign, then f has at least one
zero between a and b.
Intermediate Value Theorem

Example.
Problem: Show that
f(x) = x5 { x4 + 7x3 { 7x2 { 18x + 18
has a zero between 1.4 and 1.5.
Approximate it to two decimal places.
Answer:
Key Points









Division Algorithm
Remainder Theorem
Factor Theorem
Number of Real Zeros
Rational Zeros Theorem
Finding Zeros of a Polynomial
Factoring Polynomials
Bounds on Zeros
Intermediate Value Theorem
Complex Zeros;
Fundamental
Theorem of Algebra
Section 3.7
Complex Polynomial
Functions

Complex polynomial function: Function
of the form
f(x) = anxn + an {1xn {1 +  + a1x + a0






an, an {1, …, a1, a0 are all complex numbers,
an  0,
n is a nonnegative integer
x is a complex variable.
Leading coefficient of f: an
Complex zero: A complex number r with
f(r) = 0.
Complex Arithmetic



See Appendix A.6.
Imaginary unit: Number i with
i2 = {1.
Complex number: Number of the form z =
a + bi




a and b real numbers.
a is the real part of z
b is the imaginary part of z
Can add, subtract, multiply

Can also divide (we won’t)
Complex Arithmetic

Conjugate of the complex number
a + bi

Number a { bi

Written

Properties:




Complex Arithmetic

Example. Suppose z = 5 + 2i and
w = 2 { 3i.
(a) Problem: Find z + w
Answer:
(b) Problem: Find z { w
Answer:
(c) Problem: Find zw
Answer:
(d) Problem: Find
Answer:
Fundamental Theorem of
Algebra

Theorem. [Fundamental Theorem of
Algebra]
Every complex polynomial function
f(x) of degree n ¸ 1 has at least one
complex zero.
Fundamental Theorem of
Algebra

Theorem.
Every complex polynomial function f(x) of
degree n ¸ 1 can be factored into n linear
factors (not necessarily distinct) of the
form
f(x) = an(x { r1)(x { r2)  (x { rn)
where an, r1, r2, …, rn are complex
numbers. That is, every complex
polynomial function f(x) of degree n ¸ 1
has exactly n (not necessarily distinct)
zeros.
Conjugate Pairs Theorem

Theorem. [Conjugate Pairs Theorem]
Let f(x) be a polynomial whose
coefficients are real numbers. If a + bi
is a zero of f, then the complex
conjugate a { bi is also a zero of f.
Conjugate Pairs Theorem

Example. A polynomial of degree 5
whose coefficients are real numbers
has the zeros {2, {3i and 2 + 4i.
Problem: Find the remaining two zeros.
Answer:
Conjugate Pairs Theorem

Example.
Problem: Find a polynomial f of degree 4
whose coefficients are real numbers and
that has the zeros {2, 1 and 4 + i.
Answer:
Conjugate Pairs Theorem

Example.
Problem: Find the complex zeros of the
polynomial function
f(x) = x4 + 2x3 + x2 { 8x { 20
Answer:
Key Points




Complex Polynomial Functions
Complex Arithmetic
Fundamental Theorem of Algebra
Conjugate Pairs Theorem
Properties of
Rational
Functions
Section 3.3
Rational Functions

Rational function: Function of the
form
p and q are polynomials,
 q is not the zero polynomial.


Domain: Set of all real numbers
except where q(x) = 0
Rational Functions
is in lowest terms:



The polynomials p and q have no
common factors
x-intercepts of R:

Zeros of the numerator p when R is in
lowest terms
Rational Functions

Example. For the rational function
(a) Problem: Find the domain
Answer:
(b) Problem: Find the x-intercepts
Answer:
(c) Problem: Find the y-intercepts
Answer:
Graphing Rational
Functions

Graph of
10
7.5
5
2.5
-10
-5
5
-2.5
-5
-7.5
-10
10
Graphing Rational Functions

As x approaches 0,
is unbounded in the positive
direction.
Write f(x) ! 1
 Read “f(x) approaches infinity”
 Also:

May write f(x) ! 1 as x ! 0
 May read: “f(x) approaches infinity as x
approaches 0”

Graphing Rational Functions

Example. For
Problem: Use transformations to graph f.
Answer:
4
2
-6
-4
-2
2
-2
-4
4
Asymptotes

Horizontal asymptotes:
Let R denote a function.
 Let x ! {1 or as x ! 1,
 If the values of R(x) approach some
fixed number L, then the line y = L is a
horizontal asymptote of the graph of R.

Asymptotes

Vertical asymptotes:
Let x ! c
 If the values jR(x)j ! 1, then the line
x = c is a vertical asymptote of the
graph of R.

Asymptotes

Asymptotes:


Oblique asymptote: Neither horizontal
nor vertical
Graphs and asymptotes:
Graph of R never intersects a vertical
asymptote.
 Graph of R can intersect a horizontal or
oblique asymptote (but doesn’t have to)

Asymptotes

A rational function can have:
Any number of vertical asymptotes.
 1 horizontal and 0 oblique asymptote
 0 horizontal and 1 oblique asymptotes
 0 horizontal and 0 oblique asymptotes
 There are no other possibilities

Vertical Asymptotes

Theorem. [Locating Vertical
Asymptotes]
A rational function
in lowest terms, will have a vertical
asymptote x = r if r is a real zero of
the denominator q.
Vertical Asymptotes

Example. Find the vertical
asymptotes, if any, of the graph of
each rational function.
(a) Problem:
Answer:
(b) Problem:
Answer:
Vertical Asymptotes

Example. (cont.)
(c) Problem:
Answer:
(d) Problem:
Answer:
Horizontal and Oblique
Asymptotes


Describe the end behavior of a
rational function.
Proper rational function:


Degree of the numerator is less than the
degree of the denominator.
Theorem.
If a rational function R(x) is proper,
then y = 0 is a horizontal asymptote
of its graph.
Horizontal and Oblique
Asymptotes

Improper rational function R(x): one
that is not proper.

May be written
where
is proper. (Long division!)
Horizontal and Oblique
Asymptotes

If f(x) = b, (a constant)


If f(x) = ax + b, a  0,


Line y = b is a horizontal asymptote
Line y = ax + b is an oblique asymptote
In all other cases, the graph of R
approaches the graph of f, and there
are no horizontal or oblique
asymptotes.

This is all higher-degree polynomials
Horizontal and Oblique
Asymptotes

Example. Find the hoizontal or
oblique asymptotes, if any, of the
graph of each rational function.
(a) Problem:
Answer:
(b) Problem:
Answer:
Horizontal and Oblique
Asymptotes

Example. (cont.)
(c) Problem:
Answer:
(d) Problem:
Answer:
Key Points




Rational Functions
Graphing Rational Functions
Vertical Asymptotes
Horizontal and Oblique Asymptotes
The Graph of a
Rational Function;
Inverse and Joint
Variation
Section 3.4
Analyzing Rational Functions



Find the domain of the rational
function.
Write R in lowest terms.
Locate the intercepts of the graph.
x-intercepts: Zeros of numerator of
function in lowest terms.
 y-intercept: R(0), if 0 is in the domain.


Test for symmetry – Even, odd or
neither.
Analyzing Rational Functions

Locate the vertical asymptotes:




Zeros of denominator of function in
lowest terms.
Locate horizontal or oblique
asymptotes
Graph R using a graphing utility.
Use the results obtained to graph by
hand
Analyzing Rational Functions

Example.
Problem: Analyze the graph of the
rational function
Answer:
Domain:
R in lowest terms:
x-intercepts:
y-intercept:
Symmetry:
Analyzing Rational Functions

Example. (cont.)
Answer: (cont.)
Vertical asymptotes:
Horizontal asymptote:
Oblique asymptote:
Analyzing Rational Functions

Example. (cont.)
Answer: (cont.)
4
2
-4
-2
2
-2
-4
4
Analyzing Rational Functions

Example.
Problem: Analyze the graph of the
rational function
Answer:
Domain:
R in lowest terms:
x-intercepts:
y-intercept:
Symmetry:
Analyzing Rational Functions

Example. (cont.)
Answer: (cont.)
Vertical asymptotes:
Horizontal asymptote:
Oblique asymptote:
Analyzing Rational Functions

Example. (cont.)
Answer: (cont.)
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Variation

Inverse variation:
Let x and y denote 2 quantities.
 y varies inversely with x



If there is a nonzero constant such that
Also say: y is inversely proportional to
x
Variation

Joint or Combined Variation:
Variable quantity Q proportional to the
product of two or more other variables
 Say Q varies jointly with these
quantities.
 Combinations of direct and/or inverse
variation are combined variation.

Variation

Example. Boyle’s law states that for a
fixed amount of gas kept at a fixed
temperature, the pressure P and
volume V are inversely proportional
(while one increases, the other
decreases).
Variation

Example. According to Newton, the
gravitational force between two
objects varies jointly with the
masses m1 and m2 of each object
and inversely with the square of the
distance r between the objects,
hence
Key Points


Analyzing Rational Functions
Variation
Polynomial and
Rational
Inequalities
Section 3.5
Solving Inequalities
Algebraically

Rewrite the inequality
Left side: Polynomial or rational
expression f. (Write rational expression
as a single quotient)
 Right side: Zero
 Should have one of following forms

f(x)
 f(x)
 f(x)
 f(x)

>
¸
<
·
0
0
0
0
Solving Inequalities
Algebraically


Determine where left side is 0 or
undefined.
Separate the real line into intervals
based on answers to previous step.
Solving Inequalities
Algebraically

Test Points:
Select a number in each interval
 Evaluate f at that number.
 If the value of f is positive, then
f(x) > 0 for all numbers x in the
interval.
 If the value of f is negative, then
f(x) < 0 for all numbers x in the
interval.

Solving Inequalities
Algebraically

Test Points (cont.)

If the inequality is strict (< or >)
Don’t include values where x = 0
 Don’t include values where x is undefined.


If the inequality is not strict (· or ¸)
Include values where x = 0
 Don’t include values where x is undefined.

Solving Inequalities
Algebraically

Example.
Problem: Solve the inequality x5 ¸ 16x
Answer:
Key Points

Solving Inequalities Algebraically