polydiv on the TI-86

The polydiv program for the TI-85 and TI-86 does division of polynomials, giving not only the quotient and the remainder, but also all of the steps along the way. (The program can be downloaded for the TI-85 as polydiv.85p and for the TI-86 as polydiv.85p.) In order to do this, the TI-86 uses a matrix to hold the divisor, the dividend, the quotient, the remainder, and all of the rest of the computations. For example consider the problem on the left, and its representation in polydiv on the right.

As we can see from the illustration above, the polydiv program produces all of the numbers associated with the traditional polynomial division algorithm. This page will demonstrate the use of the polydiv program on the TI-88 and TI-86. We will start by doing the problem illustrated above. In that problem we have the DIVISOR 1x + 8 and the DIVIDEND 1x² + 7x – 9 We will need to represent the coefficients of these polynomials to the program. The polydiv program uses vectors to hold the coefficients. Vectors were chosen because the symbols for creating vectors, namely, [ and ], are on the keyboard.

Figure 1 Figure 1 is the result of pressing the key to open the PROGRAM menu, using the and then key if need be to shift the display to the point where the "polydi" option is visible, and then using the appropriate key to select the desired option. In the case shown in Figure 1 we use the key to paste the name of the program onto the screen. Once the polydiv program has been pasted onto the screen, we press the key to move to Figure 2.

Figure 2 Figure 2 shows the program window after we havee responded to the two prompts that the program gives us. First, the program asks us for the DIVISOR VECTOR. We respond with [1,8] and then we press the key to submit that response. The program prompts us for the DIVIDEND VECTOR. For our problem we respond with [1,7,^–9] To submit that response and move to Figure 3 we press the key.

Figure 3 The polydiv program displays the initial matrix in Figure 3. At this point the initial division has been done. That is, the program reflects the problem at the stage of completing the initial division, namely, The program is in a "paused" condition to allow us to inspect the matrix of values. We press to continue the division and move to Figure 4.

Figure 4 In Figure 4 the program has taken another step, determining the second coefficient in the quotient. In this case, that is really all that there is to the problem. Again the program is in a paused state so that we can move around in the matrix if need be. For this problem we can see all of the answer on the screen. Therefore, we do not need to look at any hidden values. We press to continue, and move to Figure 5.

Figure 5 Once the program has reached the end of the division, the polydiv program displays the menu shown in Figure 5. This allows us to
start a new problem by pressing 0
end the POLYDIV1 program by pressing 1
show the answer to the last problem by pressing 2.

Let us look at another problem, namely,

Here the DIVIDEND VECTOR will be [1,2,0,^–5] with [1,3] as the DIVISOR VECTOR.

Figure 6 We left Figure 5 by pressing the key to tell the calculator that we have a new problem to do. The polydiv program again asks for the DIVISOR VECTOR, to which we respond with [1,3] and then . The program then asks for the DIVIDEND VECTOR. Figure 6 shows that vector as [1,2,0,^–5]

Figure 7 Pressing moves the program from Figure 6 to Figure 7. The program is paused. We can examine the output and note that the first coefficient of the quotient is 1. Press to move to Figure 8.

Figure 8 Figure 8 shows the calculator display after the second cycle of the division algorithm. The second coefficient of the quotient has been computed to be "^–1" and the corresponding portion of the division algorithm has been completed. Press to move to Figure 9.

Figure 9 In the earlier Figures we could see all that we needed to see on the screen. Now, in Figure 9, we can see some of the values but we are missing the bottom numbers. Because the program is paused, we can use the cursor keys to move around on the screen. For example, we can press the key to move down, as seen in Figure 10.

Figure 10 Figure 10 displays the bottom of the matrix. Now that we can see the bottom of the matrix we note that the problem is done. If we press the ENTER the program will display the options to do a new problem, quit, or re-display the current matrix.

Here is one more example:

Now the DIVIDEND VECTOR will be [14,^–47,14,58,^–49] with [2,^–5] as the DIVISOR VECTOR.

Figure 11 In Figure 11 we have pressed to ask for a new problem, and we have entered both the DIVISOR LIST and the DIVIDEND LIST.

Figure 12 Press to move from Figure 11 to Figure 12. Here we can see the matrix after the first step of the algorithm. Unfortunately, we can only see the left side of the matrix. We press the key to move the display to the right so that we can see the rest of the DIVIDEND in Figure 13.

Figure 13 Having seen the entire DIVIDEND, we press to continue the program.

Figure 14 Figure 14 shows the matrix after the second step of the division algorithm. The second coefficient of the quotient has been computed to be "^–6".

Figure 15 To move from Figure 14 to Figure 15 we press to take the next step in the algorithm.

Figure 16 Then, in Figure 16, we use the key to shift the display to the right and the key to shift the display down. This allows us to look at all of the matrix. There is more to do for the problem. Therefore, press the key to move to Figure 17.

Figure 17 At this point all of the updates from the program are happenning in matrix cells that are off the screen. We can use the cursor keys to move the display around so that we can see more of the matrix.

Figure 18 Figure 18 shows the bottom right side of the matrix. We press one more time. This will display the menu shown in Figure 19.

Figure 19 Here we chose option 1 to end the program. Had we chosen option 2 the program would have re-displayed the matrix so that we could rove around it to see all of the values.

Figure 1	Figure 1 is the result of pressing the key to open the PROGRAM menu, using the and then key if need be to shift the display to the point where the "polydi" option is visible, and then using the appropriate key to select the desired option. In the case shown in Figure 1 we use the key to paste the name of the program onto the screen. Once the polydiv program has been pasted onto the screen, we press the key to move to Figure 2.
Figure 2	Figure 2 shows the program window after we havee responded to the two prompts that the program gives us. First, the program asks us for the DIVISOR VECTOR. We respond with [1,8] and then we press the key to submit that response. The program prompts us for the DIVIDEND VECTOR. For our problem we respond with [1,7,^–9] To submit that response and move to Figure 3 we press the key.
Figure 3	The polydiv program displays the initial matrix in Figure 3. At this point the initial division has been done. That is, the program reflects the problem at the stage of completing the initial division, namely, The program is in a "paused" condition to allow us to inspect the matrix of values. We press to continue the division and move to Figure 4.
Figure 4	In Figure 4 the program has taken another step, determining the second coefficient in the quotient. In this case, that is really all that there is to the problem. Again the program is in a paused state so that we can move around in the matrix if need be. For this problem we can see all of the answer on the screen. Therefore, we do not need to look at any hidden values. We press to continue, and move to Figure 5.
Figure 5	Once the program has reached the end of the division, the polydiv program displays the menu shown in Figure 5. This allows us to start a new problem by pressing 0 end the POLYDIV1 program by pressing 1 show the answer to the last problem by pressing 2.

Figure 6	We left Figure 5 by pressing the key to tell the calculator that we have a new problem to do. The polydiv program again asks for the DIVISOR VECTOR, to which we respond with [1,3] and then . The program then asks for the DIVIDEND VECTOR. Figure 6 shows that vector as [1,2,0,^–5]
Figure 7	Pressing moves the program from Figure 6 to Figure 7. The program is paused. We can examine the output and note that the first coefficient of the quotient is 1. Press to move to Figure 8.
Figure 8	Figure 8 shows the calculator display after the second cycle of the division algorithm. The second coefficient of the quotient has been computed to be "^–1" and the corresponding portion of the division algorithm has been completed. Press to move to Figure 9.
Figure 9	In the earlier Figures we could see all that we needed to see on the screen. Now, in Figure 9, we can see some of the values but we are missing the bottom numbers. Because the program is paused, we can use the cursor keys to move around on the screen. For example, we can press the key to move down, as seen in Figure 10.
Figure 10	Figure 10 displays the bottom of the matrix. Now that we can see the bottom of the matrix we note that the problem is done. If we press the ENTER the program will display the options to do a new problem, quit, or re-display the current matrix.

Figure 11	In Figure 11 we have pressed to ask for a new problem, and we have entered both the DIVISOR LIST and the DIVIDEND LIST.
Figure 12	Press to move from Figure 11 to Figure 12. Here we can see the matrix after the first step of the algorithm. Unfortunately, we can only see the left side of the matrix. We press the key to move the display to the right so that we can see the rest of the DIVIDEND in Figure 13.
Figure 13	Having seen the entire DIVIDEND, we press to continue the program.
Figure 14	Figure 14 shows the matrix after the second step of the division algorithm. The second coefficient of the quotient has been computed to be "^–6".
Figure 15	To move from Figure 14 to Figure 15 we press to take the next step in the algorithm.
Figure 16	Then, in Figure 16, we use the key to shift the display to the right and the key to shift the display down. This allows us to look at all of the matrix. There is more to do for the problem. Therefore, press the key to move to Figure 17.
Figure 17	At this point all of the updates from the program are happenning in matrix cells that are off the screen. We can use the cursor keys to move the display around so that we can see more of the matrix.
Figure 18	Figure 18 shows the bottom right side of the matrix. We press one more time. This will display the menu shown in Figure 19.
Figure 19	Here we chose option 1 to end the program. Had we chosen option 2 the program would have re-displayed the matrix so that we could rove around it to see all of the values.