Use synthetic division to divide the first polynomial by the second.
x³ - 2x + 1
x + 4
Write the first polynomial as
x³ + 0x² - 2x + 1
Change the sign of +4 to -4
-4|1 0 -2 1
|
Bring down the 1
-4|1 0 -2 1
|
1
Multiply the 1 by the -4 and write it above above and to the right
of the 1 that you just brought down
-4|1 0 -2 1
| -4
1
Add the 0 and the -4, getting -4, and write it under the -4
-4|1 0 -2 1
| -4
1 -4
Multiply the -4 at the bottom by the -4 at the far left, getting 16.
Write that above and to the right of the -4 at the bottom:
-4|1 0 -2 1
| -4 16
1 -4
Add the -2 and the 16, getting 14, and write it under the 16
-4|1 0 -2 1
| -4 16
1 -4 14
Multiply the 14 at the bottom by the -4 at the far left, getting -56.
Write that above and to the right of the 14 at the bottom:
-4|1 0 -2 1
| -4 16 -56
1 -4 14
Add the 1 and the -56, getting -55 and write that under the -56
-4|1 0 -2 1
| -4 16 -56
1 -4 14 -55
Now we must interpret the numbers at the bottom. The last number
-55 is the remainder, and the numbers to the left of it are the
coefficients of the quotient polynomial which has degree which is
one less than the original polynomial:
So the answer is
-55
x² - 4x + 14 + ------
x+4
Or if you prefer:
55
x² - 4x + 14 - ------
x+4
The synthetic division is just a shortcut for this long division:
x² - 4x + 14
x + 4)x³ + 0x² - 2x + 1
x³ + 4x²
-4x² - 2x
-4x² - 16x
14x + 1
14x + 56
-55
which yields the same answer.
Edwin
