5.6: Find Rational Zeros, II

5.6: Find Rational Zeros, II
Objectives:
1. To use a graph to
help find the zeros
of a polynomial
function
2. To use the Location
Principle to help find
the zeros of a
polynomial function
Assignment:
• P. 374-377: 2, 19-23,
24-34 even, 41-44,
48, 49, 53
• Challenge Problems:
1-2
Warm-Up
The points (2, −3)
and (3, 5) are on
the graph of a
polynomial
function f (x).
What conclusion
can you draw
about the graph
of f (x)?
There must
be a zero
between 2
and 3 since
the graph
must cross
the 𝑥-axis to
connect
these points.
The Rational Zero Theorem
If f(x) = anxn + … + a1x + a0 has integer
coefficients, then every rational zero of f(x)
has the following form:
𝑝
factor of constant term 𝑎0
=
𝑞 factor of leading coefficient 𝑎𝑛
Important note: These factors can be either
positive or negative.
The Rational Zero Theorem
Here’s another way to think about The
Rational Zero Test. Consider the function
below:
𝑓 𝑥 = 6𝑥 3 + 𝑥 2 − 47𝑥 − 30
In factored form:
𝑓 𝑥 = 2𝑥 + 5 𝑥 − 3 3𝑥 + 2
And here are the zeros:
5
2
𝑥 = − , 3, and −
2
3
The Rational Zero Theorem
Now work backwards from the factors: What
would be the constant term and the leading
coefficient?
𝑓 𝑥 = 6𝑥 3 + 𝑥 2 − 47𝑥 − 30
𝑓 𝑥 = 2𝑥 + 5 𝑥 − 3 3𝑥 + 2
(2 x)( x)(3x)  6 x3
Factors of 6
5
2
𝑥 = − , 3, and −
2
3
(5)(3)(2)  30
Factors of −30
The Rational Zero Theorem
Remember that The Rational Zero Theorem
just helps you to generate a list of the
possible rational zeros of a polynomial
function (with integer coefficients).
To verify each zero, you have to use
synthetic division.
Exercise 1
List the possible rational zeros of f(x) using the
Rational Zero Theorem.
𝑓 𝑥 = 𝑥 3 − 12𝑥 2 + 35𝑥 − 24
Exercise 2
Find all real zeros of the function.
𝑓 𝑥 = 𝑥 3 − 12𝑥 2 + 35𝑥 − 24
Exercise 3
List the possible rational zeros of f(x) using
the Rational Zero Theorem.
𝑓 𝑥 = 2𝑥 3 + 5𝑥 2 − 11𝑥 − 14
Exercise 3
List the possible rational zeros of f(x) using
the Rational Zero Theorem.
𝑓 𝑥 = 2𝑥 3 + 5𝑥 2 − 11𝑥 − 14
Notice that when the leading coefficient is not 1,
the number of possible rational zeros
increases in an unsavory way. Surely there
has to be a way to narrow the possibilities…
Exercise 4
According to The Rational Zero Theorem,
what is the maximum number of possible
rational zeros for the polynomial function
below?
𝑓 𝑥 = 𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑
Exercise 5
Assume a has k factors and d has r factors.
According to The Rational Zero Theorem,
what is the maximum number of possible
rational zeros for the polynomial function
below?
𝑓 𝑥 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑
Objective 1
You will be able to use a graph to help
find the zeros of a polynomial function
Facilitation, 1 (Cheating)
The first way to
facilitate the search
for rational zeros
involves looking at
the graph of the
polynomial.
Facilitation, 1 (Cheating)
Since zeros are the same
thing as a graph’s xintercepts, finding the
approximate location of
an x-intercept can give a
reasonable place to
start verifying factors
from our long, tedious
list of possibilities.
Exercise 6
Find all real zeros of the function.
𝑓 𝑥 = 2𝑥 3 + 5𝑥 2 − 11𝑥 − 14
Facilitation, 1 (Cheating)
Using the graph of a polynomial
function to help finding its zeros
seems a bit like cheating. This
is because we want to find the
zeros (or x-intercepts), and we
do this by looking a graph’s xintercepts. If we already have a
graph, then why are we trying
to find the graph’s x-intercepts
algebraically?
(A fine example
of circular logic)
Exercise 7
Find all real zeros of the function.
𝑓 𝑥 = 48𝑥 3 + 4𝑥 2 − 20𝑥 + 3
Exercise 8
Find all real zeros of the function.
𝑓 𝑥 = 2𝑥 4 + 5𝑥 3 − 18𝑥 2 − 19𝑥 + 42
Objective 2
You will be
able to use
the Location
Principle to
help find the
zeros of a
polynomial
function
Location Principle
Another way to narrow
𝑥
𝑓(𝑥)
down our rational zero
0
−6
choices is by using the
1
−12
Location Principle.
2
28
Consider the table of
3
150
values. What can you
4
390
conclude about the
5
784
location of one of the
There must be a zero here
zeros of f(x)?
Location Principle
Location Principle
If 𝑓 is a polynomial
function and 𝑎 and
𝑏 are two numbers
such that 𝑓 𝑎 < 0
and 𝑓 𝑏 > 0, then
𝑓 has at least one
zero between 𝑎
and 𝑏.
For example, if
𝑓(2) = −3 and
𝑓(3) = 5, then there
must be a zero between
𝑥 = 2 and 𝑥 = 3. This is
because a continuous
curve would cross the 𝑥axis when connecting
these points.
Location Principle
Location Principle
If 𝑓 is a polynomial
function and 𝑎 and
𝑏 are two numbers
such that 𝑓 𝑎 < 0
and 𝑓 𝑏 > 0, then
𝑓 has at least one
zero between 𝑎
and 𝑏.
There must
be a zero
between 2
and 3 since
the graph
must cross
the 𝑥-axis
to connect
these
points.
Location Principle
Location Principle
If 𝑓 is a polynomial
function and 𝑎 and
𝑏 are two numbers
such that 𝑓 𝑎 < 0
and 𝑓 𝑏 > 0, then
𝑓 has at least one
zero between 𝑎
and 𝑏.
So we could use a
table of values with
the Location Principle
to narrow our rational
zero choices. Just
look for where the
function-values
change signs: + to –
or – to +.
Exercise 9
Find all real zeros of the function.
𝑓 𝑥 = 9𝑥 4 + 3𝑥 3 − 30𝑥 2 + 6𝑥 + 12
𝑥
𝑓(𝑥)
-3
372
-2
0
-1
-18
0
12
1
0
2
72
3
570
Exercise 9
Find all real zeros of the function.
𝑓 𝑥 = 9𝑥 4 + 3𝑥 3 − 30𝑥 2 + 6𝑥 + 12
𝑥
𝑓(𝑥)
-3
372
-2
0
-1
-18
0
12
1
0
2
72
3
570
Exercise 9
Find all real zeros of the function.
𝑓 𝑥 = 9𝑥 4 + 3𝑥 3 − 30𝑥 2 + 6𝑥 + 12
𝑥
𝑓(𝑥)
-3
372
-2
0
-1
-18
0
12
1
0
2
72
3
570
Exercise 10
The Rational Zero Theorem only helps in
finding rational zeros. How would we go
about finding the irrational zeros of a
polynomial function?
Once it comes down to an unfactorable
quadratic, use the quadratic formula or
completing the square to find the last two
zeros.
Exercise 11
Find all real zeros of the function.
𝑓 𝑥 = 24𝑥 4 − 38𝑥 3 − 191𝑥 2 − 157𝑥 − 28
𝑥
𝑓(𝑥)
-2
210
-1
0
0
-28
1
-390
2
-1026
3
-1300
4
0
Exercise 12
Find all real zeros of the function.
𝑓 𝑥 = 10𝑥 4 − 11𝑥 3 − 42𝑥 2 + 7𝑥 + 12
𝑥
𝑓(𝑥)
-3
720
-2
78
-1
-16
0
12
1
-24
2
-70
3
168
5.6: Find Rational Zeros, II
Objectives:
1. To use a graph to
help find the zeros
of a polynomial
function
2. To use the Location
Principle to help find
the zeros of a
polynomial function
Assignment
• P. 374-377: 2, 1923, 24-34 even,
41-44, 48, 49, 53
• Challenge
Problems: 1-2
“Which came first: the graph or the zeros?”