Texas Instruments TI-83 Plus Silver Edition Manual

7/31/01 ©2001 Texas Instruments
TI-83 Plus Polynomial Root
Finder & Simultaneous
Equation Solver Application
Getting Started
Polynomial Root Finder – Precalculus Level Example
Polynomial Root Finder – Calculus Level Example
Simultaneous Equation Solver
How To…
Load Values from a List Load Values from a Matrix
Store Data into Lists Store Data into Matrices
Examples
Finding the Roots of a Polynomial
Solving a System of Equations
Displaying a Matrix in Reduced Row-Echelon Form
More Information
Customer Support Error Recovery
TI

TI-83 Plus Polynomial Root Finder / Simultaneous Equation Solver Application Page 2
Important Information
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TI-83 Plus Polynomial Root Finder / Simultaneous Equation Solver Application Page 3
What Is the Polynomial Root Finder &
Simultaneous Equation Solver Application?
The Polynomial Root Finder application calculates the roots
(zeros) of polynomials of degree 1 through 30, with a convenient,
easy to use interface. You can store solutions into lists, load lists
into the application for editing, and graph the polynomial.
Because the application uses the operating system’s eigenvalue
routines, solutions are computed numerically only.
The Simultaneous Equation Solver application finds solutions to
systems of linear equations. You can load matrices containing
the coefficients of linear systems and identify whether a given
system has a unique solution, an infinite number of solutions, or
no solution.

TI-83 Plus Polynomial Root Finder / Simultaneous Equation Solver Application Page 4
What You Need to Install and Use the Application
To install and run the application, you need:
•A TI-83 Plus calculator with version 1.13 or later of the
operating system software to optimize the performance of your
calculator and the application.
-To check the operating system version, press \/,
and then select About. The version number is displayed
below the product name.
-You can download a free copy of the latest operating
system software from education.ti.com/softwareupdates.
Follow the link to Operating Systems.
•A computer with WindowsÍ95/98/2000, Windows NTÍ, or
Apple® Mac® OS 7.0 or later installed.
•A TI-GRAPH LINK™ computer-to-calculator cable. If you do
not have this cable, call your distributor, or order the cable
online from TI's online store.
•TI-GRAPH LINK software that is compatible with the
TI83 Plus. You can download a free copy of the
TI-GRAPH LINK software from
education.ti.com/softwareupdates. Follow the link to
Connectivity Software, then TI-GRAPH LINK.

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Where to Find Installation Instructions
Detailed installation instructions are available from
education.ti.com/guides. Follow the link to Flash installation
instructions.
Getting Help
This application contains built-in help screens that give you
information about how to use both the Polynomial Root Finder
and the Simultaneous Equation Solver applications. To access
and use the help screens:
1. From the
MAIN MENU, select either Poly Help or Simult Help.
2. Press
ato display subsequent screens.
3. Press
\to exit the help screens and return to the
MAIN MENU.
Press \to
exit the help
screens and return
to the Main Menu.
Press ato view
the next help
screen.

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Polynomial Root Finder Application
Getting Started with the Polynomial Root Finder Application
Precalculus Level Example
Work through the following example to get acquainted with the
main features of the Polynomial Root Finder application.
A golfer hits a golf ball from a tee at the bottom of a hill. The
hill can be described by y(x) = .8x. The ball follows the path
y(x) = 0x2+12x. If the golfer is standing at the tee (0,0) and
hits the ball, where does the ball hit the ground on the hill?
To solve this problem, you need to find where the path of the
golf ball, y(x) = 0x2+12x, will hit the hill, y(x) = .8x. Solve this
by hand first, and then follow the instructions below to verify
your calculation.
hill
path of the
golf ball

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1. First, equate the two equations.
0x2+12x = .8x
0x2+11.2x = 0
The roots of the resulting polynomial will give the intersection
points of the ball path and the hill.
2. Press
nto display a list of applications on your calculator.
3. Select
PolySmlt. The information screen is displayed.
4. Press any key to continue. The MAIN MENU is displayed.
5. Select
Poly Root Finder.
6. Enter the degree of the polynomial (2), and then press ¯.
7. Enter the coefficients of the polynomial {01, 11.2, 0}. Press
¯after each coefficient to move the cursor to the next
line.

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8. Select
SOLVE (press V) to calculate and display the
roots.
Two answers, x1=11.2 and x2=0, are displayed. If you exit the
application, graph the two functions, and find their intersection
(press \, and then select Intersect), you can see that the
ball starts at (0,0) and hits the hill at (11.2, 8.96).
To see this graph:
1. Press T
2. Select Zoom Out.
3. Move the cursor near (0,0).
4. Press ¯.

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Calculus Level Example
Work through the following example to get acquainted with the
main features of the Polynomial Root Finder application.
An open box is to be made from a square piece of material 36
centimeters on a side by cutting equal squares from the
corners and turning up the sides. What is the largest volume
box that can be made with this construction?
L = (36 – 2x)
W=(36 – 2x)
H=x
The volume of the box is V(x) = x(36-2x)2.
36 cm
x
36 –2x cm

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1. To find the extreme values of the volume, set the first
derivative of V to zero.
V(x) = x(36-2x)2
V(x) = 4x3–144x2+1296x
V'(x) = 12x2–288x + 1296 = 0
x2–24x + 108 = 0
2. Now, if you find the roots of x2–24x + 108 = 0, you will find
the critical points of the function.
3. Press
nto display a list of applications on your calculator.
4.Select PolySmlt. The information screen is displayed.
5. Press any key to continue. The MAIN MENU is displayed.
6. Select
Poly Root Finder.
7. Enter the degree of the polynomial (2), and then press ¯.

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8. Enter the coefficients of the polynomial {1, /24, 108}. Press
¯after each coefficient to move the cursor to the next
line.
9. Select
SOLVE (press V) to calculate and display the
roots.
Two answers, x1=18 and x2=6, are displayed. Which do you
choose?

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10. To test that x = 6 is the maximum, remember to check the
sign of the second derivative at x = 6 (i.e., make sure that
0
(6)V
<
′
′
).
If you graph V(x), you can see that 6 is the maximum value for x
that you can use to construct a box.
11. Since the volume of the box is V(x) = x(36-2x)2, you can find
the maximum volume by substituting 6 for x. (Answer: 3456)
Now that you have completed the Getting Started example, you
can read further for more detailed instructions on how to use the
Polynomial Root Finder application. Some topics you might want
to explore are:
•Loading a list of coefficients into the application
•Storing coefficients to a list
•Storing solutions (roots) to a list
•Storing equations to the Y= editor
The reasonable
physical domain has
x from 0 to 18 (fold
the paper in half).
This window gives a
nice picture of the
function for positive
x = 0 to 25.

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Starting and Quitting the Polynomial Root Finder Application
Starting the Application
1. Press
nto display a list of applications on your calculator.
2. Select
PolySmlt. The information screen is displayed.
3. Press any key to display the Main Menu.
4. Select
Poly Root Finder. The POLY ROOT FINDER main screen
is displayed.
Quitting the Application
1. From any screen, select MAIN to return to the Main Menu.
2. Select
Quit PolySmlt.
Shortcut
You can also press \to exit the application. If the cursor
is at a name prompt, you must press \twice.

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Selecting Polynomial Root Finder Options
Several that you can use to accomplish specific tasks options
are displayed across the bottom of each screen. To select an
option, press the graphing key directly under the option.
Options You Can Use
Use options on this screen …
To do this:
Degree entry screen
MAIN displays the main menu.
LOAD displays a list prompt so that you
can enter a list name that contains the
coefficient values you want to use. You
must set up the list in the list editor before
you can use the list in the Polynomial Root
Finder application. You cannot access the
list editor while the application is running.
Press a graphing
key to select one
of these options.
RSTUV

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Use options on this screen …
To do this:
Coefficient entry screen
MAIN displays the main menu.
DEGR displays the degree entry screen
where you can view or change the degree
of the polynomial.
CLR clears all of the coefficient values that
you have entered.
LOAD displays a list prompt so that you
can enter a list name that contains the
coefficient values you want to use. You
must set up the list in the list editor before
you can use the list in the Polynomial Root
Finder application. You cannot access the
list editor while the application is running.
SOLVE computes and displays the roots of
the polynomial.
The solution may be too long to display on
one screen. If an up or down arrow is
displayed on the left side of the screen,
press hand `as necessary to view the
entire solution.

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Use options on this screen …
To do this:
Solution screen
MAIN displays the main menu.
COEFS displays the coefficient entry
screen so that you can view or change the
coefficients.
STOa displays a list prompt so that you can
enter the list name where you want the
coefficients stored.
STOx displays a list prompt so that you can
enter the list name where you want the
solutions stored.
STOy stores the polynomial equation to the
next available y-variable in the Y= editor.

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Changing Mode Settings
This application has a unique mode settings screen that is based
on the calculator’s mode settings. These mode settings only
apply while the Polynomial Root Finder/Simultaneous Equation
Solver application is running. The settings are saved and are
applied each time you run the application. When you exit the
application, your calculator's previous mode settings are
restored.
Note
•If RAM is reset, the saved Polynomial Root Finder /
Simultaneous Equation Solver mode settings will be lost.
•Mode settings are stored in an application variable (AppVar). If
you want to share the application and your mode settings with
another TI83 Plus user, you must send both the application
and the AppVar named APPVARO.
•If a list name prompt (for example, STOx LIST = ) is displayed,
you must exit the prompt before you can change the mode
settings.
To change the mode settings:
1. Start the Polynomial Root Finder application.

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2. From any screen, press ]to display the PolySmlt MODE
Settings screen.
3. Move the cursor to a setting and press ¯to select the
setting.
Setting
Description
Normal Sci Eng
Normal
notation mode is the usual way to
express numbers, with digits to the left
and right of the decimal, as in 12345.67.
Sci (scientific) notation mode expresses
numbers in two parts. The significant
digits display with one digit to the left of
the decimal. The appropriate power of 10
displays to the right of E, as in
1.234567E4.
Eng (engineering) notation mode is
similar to scientific notation. However, the
number can have up to three digits before
the decimal. The power-of-10 exponent is
a multiple of three, as in 12.34567E3.

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Setting
Description
Float 0 1 2 3 4 5 6 7 8 9
Float (floating) decimal mode displays up
to 10 digits, plus the sign and decimal.
0123456789 (fixed) decimal mode
specifies the number of digits
(0 through 9) to display to the right of the
decimal.
Radian Degree
Radian mode interprets angle values as
radians. Answers display in radians.
Degree mode interprets angle values as
degrees. Answers display in degrees.
Real a+bi re^Ti
Real mode does not display complex
results. If you select this mode setting and
the answer is complex, NONREAL is
displayed as the result.
a+bi(rectangular complex) mode
displays complex numbers in the form
a+bi. You may need to press ato display
all of the complex number.
re^TTi(polar complex) mode displays
complex numbers in the form of reTi. You
may need to press ato display all of the
complex number.

TI-83 Plus Polynomial Root Finder / Simultaneous Equation Solver Application Page 20
4. Select
ESC to save the settings and return to the previous
screen.
Entering and Solving a Problem
To find the roots of a polynomial, you enter the degree of the
polynomial, then the value for each coefficient. The degree of the
polynomial must be an integer between 1 and 30. The value for
each coefficient must be a real number.
The following example uses the Polynomial Root Finder
application to calculate the roots of the function:
f(x) = 7x7–5x6+8x5+6x4–9x3–3x2+2x +1
1. Enter the degree of the polynomial, and then press ¯.