Intro to POLY on the TI-86 (85)

The TI-86 (and the TI-85) have a number of ways to sove polynomial equations, one of which is the "poly" key sequence,

. This page explores that function, as well as presenting some alternatives. we will be solving four problems Chapter 2.0, Example 12: 2x² + 5x – 3 = 0
Chapter 2.0, Example 13: 3x² – 8x + 5 = 0
Chapter 2.0, Example 14: 2x² – 8x + 3 = 0
and a new problem: 2x² – 5x + 8 = 0

Figure 1	We start in Figure 1 with the result of using the key sequence . The poly key command walks us through a process of solving a polynomial equation. The first thing that we need to do is to tell the calculator the "order" or "degree" of the polynomial. That is, what is the "order" or "degree" of the highest term in the polynomial?
Figure 2	The polynomial equation that we want to solve is 2x² + 5x – 3 = 0 Therefore, this is a second order polynomial, and our response to the prompt is . We finish this screen and move to Figure 3 via the key.
Figure 3	The calculator responds by showing us a standard, second order polynomial equation, namely a₂x^2+a₁x+a₀=0 and then asking for values for the three coefficients, a₂, a₁, and a₀.
Figure 4	Our equation is 2x² + 5x – 3 = 0 which means that a₂ is 2, a₁ is 5, and a₀ is ^–3. Figure 4 shows those values having been entered at the appropriate line of the display.
Figure 5	In Figure 4 we gave the calculator the values for the numeric coefficients. Now we press the key to select the fifth menu option, SOLVE. This produces the screen shown in Figure 5. This output is quite convenient in that it gives us a row of output for each of the answers to the equation.

The first five screens here show the use of the POLY key function. In Figure 5 we saw the answers for our problem. Note that the answers may need to be converted to fractional format, but this is going to be difficult with the keyboard version of the poly command.

The following three screens repeat our solution, but with the poly command. The poly command needs to be followed by a list represnting the coefficients of a polynomial. The number of items in the list determines the order of the polynomial equation. Note that it takes three items in the list to represent all three numeric coefficients in a second order polynomial equation.

Assuming that we have lots of poynomials to solve, it might be nice to put the "poly" command into our CUSTOM menu. Figures 9 through 12 step us through this process.

Figure 6 To create Figure 6 we have merely typed out the entire command. The keys shift and lock the calculator in lower case alphabetic mode. The keys produce the word poly. We leave alphabetic mode by pressing the key. We open the LIST menu by pressing the keys. This displays the menu shown in Figure 6. We can use that menu, via the key to paste aleft brace, {, onto the screen. Now we enter the values for the coefficients, separated by commas, as in amd . The selects the closing right brace, }, from the menu. With that we have constructed the poly command.

Figure 7 Having constructed the command in Figure 6, we need only press the key to perform the command, and to display the results as can be seen in Figure 7. Note that the result is given as a "list" of values. The numbers are the same as we saw in Figure 5, but there the values were displayed as separate items, one item to each line of the display.

Figure 8 One of the advantages of having the "list" of solutions is that we can obtain the values in fractional format. In Figure 8 we have used the keys to recall the previous command, the key to open the CUSTOM menu, the key to select our first menu option, , and finally, the key to perform the command that we constructed. The result is shown in Figure 8, where we can see the value of one-half being displayed as 1/2.
Figure 9 In order to put the "poly" command into the CUSTOM menu we first need to find it in the CATALOG. We do this by openning the CATALOG menu via the keys. The menu is shown in Figure 9.

Figure 10 We move forward by pressing the key to open the CATALOG. Within the CATALOG we want to find the "poly" command. We can move directly to the "P" entries by pressing the key. That will leave us with a screen as shown in Figure 10.

Figure 11 Either the cursor down key, , or the key (selecting option 1 from the menu) can be used to move down the list until we have displayed and selected the "poly" command, as shown in Figure 11. Then we need to open the CUSTOM menu, via the key.
We note in Figure 11 that there are open spots in the CUSTOM menu. We will place the "poly" command into the second spot by pressing the key. The result is shown in Figure 12.

Figure 12 Figure 12 shows us that we have placed the "poly" command into the second slot of the CUSTOM menu. We use the key a number of times to get out of the menus displayed at the bottom of the screen.

Figure 13 Having left the CATALOG menu from Figure 12, we can open the CUSTOM menu in Figure 13 by pressing the key. Then it is a simple matter, pressing the key to paste the "poly" command onto the screen.

Figure 14 Figure 14 shows the calculator display after we have completed entering the values for the equation 3x² – 8x + 5 = 0 Notice that we have openned the LIST menu so that we now have the curly braces in our menu, and that we entered the coefficients as values in a list. Having typed the command, we press the ENTER key to obtain the answer. The last output line in Figure 14 gives the decimal approximation to the answer.

Figure 15 We can change the answer to fractional form via the command, which we had stored in the CUSTOM menu. Figure 15 shows the result of returning to the CUSTOM menu via the key, and pressing the key to select the menu option. Pressing the key will produce the complete output in Figure 15.

Figure 16 In Figure 16 we are attempting to find the solutions for 2x² – 8x + 3 = 0 Notice that the two decimal approximations to the answers do not fit onto the screen. [Recall that when we used the "poly" key sequence that the solutions were listed, one per line, down the page, but that the result of the "poly" command is a list that holds the solutions.]

Figure 17 Naturally, we can use the key to see the rest of the answer from Figure 16. We have done just that to obtain the image in Figure 17.

Figure 18 Now we will look at the solution to the equation 2x² – 5x + 8 = 0 Figure 18 shows both the "poly" command and the resulting answer. In this case the answer is a pair of imaginary (complex) numbers.

Figure 19 The display for Figure 19 shows the rest of the answer from Figure 18.

Figure 20 For Figure 20 we have started the the QUAD program. (More information on the QUAD program for the TI-86 and TI-85 can be found on the 208606.htm page.) We have entered the values for the coefficients. We still need to press the key to move on to Figure 21.

Figure 21 The QUAD program calculates the discriminant and, recognizing that the discriminant is negaitve, that program merely tells us that there is no real solution.

Figure 22 QUAD1 is a more elaborate program. (More information on the QUAD1 program can be found on the 208607.htm page.) In Figure 22 we have started that program and entered the values into it.

Figure 23 Again the result is that there are no real solutions.

Figure 24 Figure 24 has the start of the QUAD2 program. (More information on the QUAD2 program can be found on the 208608.htm page.) Again we have entered the coefficients.

Figure 25 Unlike the earlier programs, QUAD2 is happy to process quadratic equations that have complex number solutions. In Figure 25 we see those two solutions, not as decimal approximations, as in Figures 18 and 19, but as complex numbers, including the square root of 39.

Figure 6	To create Figure 6 we have merely typed out the entire command. The keys shift and lock the calculator in lower case alphabetic mode. The keys produce the word poly. We leave alphabetic mode by pressing the key. We open the LIST menu by pressing the keys. This displays the menu shown in Figure 6. We can use that menu, via the key to paste aleft brace, {, onto the screen. Now we enter the values for the coefficients, separated by commas, as in amd . The selects the closing right brace, }, from the menu. With that we have constructed the poly command.
Figure 7	Having constructed the command in Figure 6, we need only press the key to perform the command, and to display the results as can be seen in Figure 7. Note that the result is given as a "list" of values. The numbers are the same as we saw in Figure 5, but there the values were displayed as separate items, one item to each line of the display.
Figure 8	One of the advantages of having the "list" of solutions is that we can obtain the values in fractional format. In Figure 8 we have used the keys to recall the previous command, the key to open the CUSTOM menu, the key to select our first menu option, , and finally, the key to perform the command that we constructed. The result is shown in Figure 8, where we can see the value of one-half being displayed as 1/2.
Figure 9	In order to put the "poly" command into the CUSTOM menu we first need to find it in the CATALOG. We do this by openning the CATALOG menu via the keys. The menu is shown in Figure 9.
Figure 10	We move forward by pressing the key to open the CATALOG. Within the CATALOG we want to find the "poly" command. We can move directly to the "P" entries by pressing the key. That will leave us with a screen as shown in Figure 10.
Figure 11	Either the cursor down key, , or the key (selecting option 1 from the menu) can be used to move down the list until we have displayed and selected the "poly" command, as shown in Figure 11. Then we need to open the CUSTOM menu, via the key. We note in Figure 11 that there are open spots in the CUSTOM menu. We will place the "poly" command into the second spot by pressing the key. The result is shown in Figure 12.
Figure 12	Figure 12 shows us that we have placed the "poly" command into the second slot of the CUSTOM menu. We use the key a number of times to get out of the menus displayed at the bottom of the screen.
Figure 13	Having left the CATALOG menu from Figure 12, we can open the CUSTOM menu in Figure 13 by pressing the key. Then it is a simple matter, pressing the key to paste the "poly" command onto the screen.
Figure 14	Figure 14 shows the calculator display after we have completed entering the values for the equation 3x² – 8x + 5 = 0 Notice that we have openned the LIST menu so that we now have the curly braces in our menu, and that we entered the coefficients as values in a list. Having typed the command, we press the ENTER key to obtain the answer. The last output line in Figure 14 gives the decimal approximation to the answer.
Figure 15	We can change the answer to fractional form via the command, which we had stored in the CUSTOM menu. Figure 15 shows the result of returning to the CUSTOM menu via the key, and pressing the key to select the menu option. Pressing the key will produce the complete output in Figure 15.
Figure 16	In Figure 16 we are attempting to find the solutions for 2x² – 8x + 3 = 0 Notice that the two decimal approximations to the answers do not fit onto the screen. [Recall that when we used the "poly" key sequence that the solutions were listed, one per line, down the page, but that the result of the "poly" command is a list that holds the solutions.]
Figure 17	Naturally, we can use the key to see the rest of the answer from Figure 16. We have done just that to obtain the image in Figure 17.
Figure 18	Now we will look at the solution to the equation 2x² – 5x + 8 = 0 Figure 18 shows both the "poly" command and the resulting answer. In this case the answer is a pair of imaginary (complex) numbers.
Figure 19	The display for Figure 19 shows the rest of the answer from Figure 18.
Figure 20	For Figure 20 we have started the the QUAD program. (More information on the QUAD program for the TI-86 and TI-85 can be found on the 208606.htm page.) We have entered the values for the coefficients. We still need to press the key to move on to Figure 21.
Figure 21	The QUAD program calculates the discriminant and, recognizing that the discriminant is negaitve, that program merely tells us that there is no real solution.
Figure 22	QUAD1 is a more elaborate program. (More information on the QUAD1 program can be found on the 208607.htm page.) In Figure 22 we have started that program and entered the values into it.
Figure 23	Again the result is that there are no real solutions.
Figure 24	Figure 24 has the start of the QUAD2 program. (More information on the QUAD2 program can be found on the 208608.htm page.) Again we have entered the coefficients.
Figure 25	Unlike the earlier programs, QUAD2 is happy to process quadratic equations that have complex number solutions. In Figure 25 we see those two solutions, not as decimal approximations, as in Figures 18 and 19, but as complex numbers, including the square root of 39.