The Location Principle
Pre-Calculus
Suppose y = f(x) represents a polynomial function
with real coefficients. If a and b are two numbers
with f(a) negative and f(b) positive, then the function
has at least one real zero between a and b.
Section 4.5
• Location Principle
• Upper Bound Theorem
• Lower Bound Theorem
The Location Principle
The Location Principle
Suppose y = f(x) represents a polynomial function
with real coefficients. If a and b are two numbers
with f(a) negative and f(b) positive, then the function
has at least one real zero between a and b.
In short if your looking at a table, watching as the x increases if
there is a sign change in the y values, if there is then there is a
zero in between the x values
Or
If you use synthetic division, and your checking a list of possible
zeros ( in some order) and the sign changes of the remainders,
when there is then there is a zero in between them.
The Location Principle
Determine between which consecutive integers the real zeros of
f(x) = x3 - 4x2 – 2x + 8 are located.
1
1
1
1
1
1
1
1
1
1
-4
-7
-6
-5
-4
-3
-2
-1
0
1
Determine between which consecutive integers the real zeros of
f(x) = 12x3 - 20x2 – x + 6 are located. Using a TABLE on calculator.
X
f(x)
-2
-168
-1
-25
0
6
1
-3
Change in sign
0 and 1, and, 1 and
2
Change in sign
2
20
3
147
So there is a zero in
between -1 and 0,
Change in sign
Try using a table on your calculator
Suppose y = f(x) represents a polynomial function with real
coefficients. If a and b are two numbers with f(a) negative and f(b)
positive, then the function has at least one real zero between a and b.
r
-3
-2
-1
0
1
2
3
4
5
Suppose y = f(x) represents a polynomial function with real
coefficients. If a and b are two numbers with f(a) negative and f(b)
positive, then the function has at least one real zero between a and b.
-2
19
10
3
-2
-5
-6
-5
-2
3
8
-49
-12
5
8
3
-4
-7
0
23
Using synthetic division
I used excell
to do this… it can
Change in sign
Be done by hand
Just as well. Just
Change in sign
Put your proposed
or synthetic division.
SHOW your tables or your synthetic divisions
Determine between which 2 consecutive integers the real
zeros exist. If they do.
f(x) = 2x2
5x + 1
p
Use q to create list of possible rational roots. To
give some clues, but remember these are only the
rational roots (not irrational or imaginary)
Roots in order
zero
Thus there is a zero at 4 and
zeros in between -2 and -1
AND between 1 and 2.
p
q
=
1
, 1
2
Answer:
0 and 1, 2 and 3
1
Looking back a the previous example. Notice when positive 5 is
checked as a root in f(x), that there are positive values in all of the
columns; thus 5 is an upper bound. Don’t bother looking at any
thing larger that 5.
Determine between which consecutive integers the real zeros of
f(x) = x3 - 4x2 – 2x + 8 are located.
Using synthetic division is an advantage here because
it is a quicker way to see the coefficients of the
depressed polynomial
Secondly it will help find a range with irrational roots.
Which the rational root theorem did not.
r
-3
-2
-1
0
1
2
3
4
5
1
1
1
1
1
1
1
1
1
1
-4
-7
-6
-5
-4
-3
-2
-1
0
1
-2
19
10
3
-2
-5
-6
-5
-2
3
8
-49
-12
5
8
3
-4
-7
0
23
Using synthetic division
Change in sign
Change in sign
zero
Thus there is a zero at 4 and zeros in
between -2 and -1 AND between 1 and 2.
Lesson Overview 4-5B
To look for the bound then you have to look at f(-x).
Since f(x) = x3 – 4x2 – 2x + 8
f(-x) = -x3 – 4x2 + 2x + 8
Looking at the syn. Divisions of f(-x) = -x3 – 4x2 + 2x + 8
r
-3
-2
-1
0
1
2
3
4
5
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-4
-1
-2
-3
-4
-5
-6
-7
-8
-9
2
5
6
5
2
-3
-10
-19
-30
-43
8
-7
-4
3
8
5
-12
-49
-112
-207
Indicates that -2 is a
lower bound for f(x)
Use the Upper and Lower Bound Theorems to find the integral
upper bound and lower bounds of
f(x) = 6x3
7x2
14x + 15
Example
The Toaster Treats Company l uses boxes with a square bottom to
package its product. The height of the box is 3 inches more than
the length of the bottom. Find the dimensions of the box to the
nearest tenth of an inch, if the volume is 42 in 3.
The Rational Root Theorem tell us that the following are possible roots
1/6, , 1/3,
½, 5/6, 1,
15/6,
3/2,
5/2, 3, 5, 15/2 , 15
Look at the nearest integers of these in synthetic division of f(x).
r
15
8
7
5
3
2
1
0
-1
-2
-3
-5
-7
-8
-15
6
-7
-14
15
Remember to use f(-x) to find
the lower bound
Answer:
Upper bound 3
Lower bound -2
V(x) = x2 (x+3)
x is the width of the bottom of the box
V(x) = x3 + 3x2
r
3
2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2
1
3
0
0
2
Example
The Toaster Treats Company l uses boxes with a square bottom to
package its product. The height of the box is 3 inches more than
the length of the bottom. Find the dimensions of the box to the
nearest tenth of an inch, if the volume is 42 in 3.
V(x) = x2 (x+3)
x is the width of the bottom of the box
V(x) = x3 + 3x2
r
3
2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2
1
1
1
1
1
1
1
1
1
1
1
3
6
5.9
5.8
5.7
5.6
5.5
5.4
5.3
5.2
5.1
0
18
17.11
16.24
15.39
14.56
13.75
12.96
12.19
11.44
10.71
0
54
49.619
45.472
41.553
37.856
34.375
31.104
28.037
25.168
22.491
2.7 inched produces the
closest value to 42 in3
So 2.7 in. by 2.7 in. by 5.7 in. box
3