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# HP 48g Series Calculators - Finding All the Roots of a Polynomial

Introduction
A polynomial is one of the most widely used functions in algebra. It is the product of constants (coefficients) and variables raised to powers (terms). Finding the roots is often referred to as “finding the zeros of a polynomial.”
An nth degree polynomial will have precisely n zeros. If f is a polynomial function of degree n > 0, then f has at least one zero in the complex number system. Remember that a polynomial function will have n zeros, they may be real or complex, and they may be repeated.
On the HP 48G calculator, the first results will be the complex (imaginary) numbers.
Calculator Symbol Key
The procedures in this document use the following text to represent symbol keys:
 Key Description Text Representation Right shift key. RS Left-shift key LS Cursor left cursor-left Cursor right cursor-right Cursor up cursor-up Cursor down cursor-down
##### note:
The menu keys are the top row of keys, with the corresponding function displayed in the boxes at the bottom of the display screen.
Instructions
To find all the roots (zeros) of a polynomial follow these steps:
1. Press the RS key, and then [SOLVE].
2. Cursor -down to highlight “Solve Poly…” and press soft menu {ok} key.
3. The COEFFICIENTS field should be highlighted.
4. Press the soft menu {edit} key.
5. Type the coefficients for the equation in the first row of the matrix.
6. Press [ENTER], this will return you to the polynomial solve menu screen.
7. Cursor -down to highlight the Roots field
8. Press the soft menu {solve} key.
9. To view the roots completely do one of the following two choices.
• Press the soft menu {edit} and highlight the first number from the left in the vector. Then, press the LS key and then the [VIEW] key. Use the arrow keys to move the cursor and fully view the roots.
• Press [ENTER]. This will place the roots on the stack. Press the arrow -up key once this will place an arrow to the left of the word “Roots”. Then press the soft menu {view} key. Use the arrow keys to move the cursor and fully view the roots.
To interpret the results, consider that the first two roots displayed are complex numbers.
Example
Given the following polynomial, find the roots:
F (x) =16x3 – 20x2 – 4x + 15
1. Press the RS key, and then press the [SOLVE] key.
2. Move the Cursor -down to the “Solve Poly…” and then press the soft menu {ok} key.
3. The Coefficients field should be highlighted.
4. Press the soft menu {edit} key.
5. Type the coefficients of the equation in the first row of the matrix.
1. 16 [ENTER]
2. 20 [+-] key [ENTER]
3. 4 [+-] key [ENTER]
4. 15 [ENTER]
6. Press [ENTER], this will return you to the solver menu screen.
7. Arrow -down to highlight the Roots field.
8. Press the soft menu {solve} key.
9. Following one of the methods in step 9 above. The results will be displayed as:
[(1, - .5) (1, .5) (- .75, 0)], this is the same answer as [- ¾, 1 ± ½ i].
The (1, -.5) and (1, .5) are imaginary numbers the (-.75, 0) is the real number.
Further Information
For more information on solving polynomials, refer to the User’s Guide on page 18-10.

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