1) The document describes performing synthetic division to divide a polynomial by a linear term (x - a). 2) It works through an example where the divisor is (x + 3), finding that a = -3. 3) It then sets up the synthetic division algorithm and carries out the steps, obtaining the quotient polynomial (3x^2 - 13x + 42) and remainder -121.

1. Synthetic Division Example

2. is the Dividend.
is the Divisor.
If the Divisor is a LINEAR TERM, in form (x –
a), you may perform synthetic division.
The first step is to determine your value of a.
So you set your divisor as equal to (x – a).

3. Finding the a value.
• x+3=x-a

4. Finding the a value.
• x+3=x–a
• -x -x
• 3 = -a

5. Finding the a value.
• x+3=x–a
• -x -x
• 3 = -a
• *-1 *-1
• -3 = a
• So, a = -3. Our first step is complete.

6. Setting up the synthetic division.
• So we have an a value of -3, we’ll put that
value down off to the left. Then we take our
dividend, , and list the
coefficients, in descending order of power.
• In this case, we have 3, -4, 3 and 5.
• If the powers are not given to us in descending
order, they must be arranged as such.
• Any skipped power of x should be entered as
having coefficient of zero.

7. Setting Up
• So we have a = -3
• Coefficients: 3, -4, 3 and 5.

8. Synthetic Division Form
• -3| 3 -4 3 5
• First we simply set this up, as such.

9. Synthetic Division Form
• -3| 3 -4 3 5
• 3
• Now we bring down the first term, 3.

10. Synthetic Division Form
• -3| 3 -4 3 5
• -9
• 3
• We multiply the 3 by our a value, -3, for -9.

11. Synthetic Division Form
• -3| 3 -4 3 5
• -9
• 3 -13
• We add -9 with the -4 above, for -13.

12. Synthetic Division Form
• -3| 3 -4 3 5
• -9 39
• 3 -13
• We multiply by a when we go diagonally up.

13. Synthetic Division Form
• -3| 3 -4 3 5
• -9 39
• 3 -13 42
• We add when we go down.

14. Synthetic Division Form
• -3| 3 -4 3 5
• -9 39 -126
• 3 -13 42
• We repeat the process.

15. Synthetic Division Form
• -3| 3 -4 3 5
• -9 39 -126
• 3 -13 42 -121
• Until we reach the final box.

16. Synthetic Division Form
• -3| 3 -4 3 5
• -9 39 -126
• 3 -13 42 -121
• Then we take our bottom line, 3, -13, 42, -121.

17. Making Sense of our Answer
• 3, -13, 42, -121.
• We started out with .
• We divided it by, among other things, x. So we
shift start with x squared, not with x cubed.
We use our bottom line, 3, -13, 42, -121 as our
coefficients.

18. Our Answer.
• 3, -13, 42, -121 becomes…
• with a remainder of -121.
• You can say that the remainder is in the form -
121 / (x + 3), since (x + 3) was the original
divisor.