There are a couple methods by which to divide polynomials such as long division, synthetic division, and more. In this story book, we’ll focus on synthetic division.

SOLVE: 3x^4 + 7x^3 + x - 11 / x+3

Step 1: Make sure that the terms in your expression are in descending order in terms of the exponent. Make sure that there are no missing terms as well!

3x^4 + 7x^3 + x - 11 -> 3x^4 + 7x^3 +0x^2 + x - 11

Now it looks good!

*Do this to make the process easier*

SOLVE: 3x^4 + 7x^3 + x - 11 / x+3

*3x^4 + 7x^3 +0x^2 + x - 11

Step 2: Write out the coefficients of each term and put the number that makes the divisor equal zero into a box to the left.

__-3| 3 7 0 1 -11

*modified equation from step 1

SOLVE: 3x^4 + 7x^3 +0x^2 + x - 11 / x+3

Step 3: Draw a line under the numbers, leaving space between them, and bring down the first number.

__-3| 3 7 0 1 -11

__________________________________

3

SOLVE: 3x^4 + 7x^3 +0x^2 + x - 11 / x+3

Step 4: Multiply this number by the number in the box, then write it under the second coefficient.

__-3| 3 7 0 1 -11

______________-9__________________

3

SOLVE: 3x^4 + 7x^3 +0x^2 + x - 11 / x+3

Step 5: Combine the product and coefficient, put the sum under the line to the right of the first number.

__-3| 3 7 0 1 -11

______________-9__________________

3 -2

SOLVE: 3x^4 + 7x^3 +0x^2 + x - 11 / x+3

Step 6: Multiply this sum by the number in the box and repeat until finished with terms.

__-3| 3 7 0 1 -11

______________-9____6___-18___51

3 -2 6 -17 40

SOLVE: 3x^4 + 7x^3 +0x^2 + x - 11 / x+3

Step 7: The numbers under the line are the coefficients of the terms in your quotient. Write them out with the first variable exponent as one less than the dividends.

__-3| 3 7 0 1 -11

______________-9____6___-18___51

3 -2 6 -17 40

3x^3 - 2x^2 + 6x -17

SOLVE: 3x^4 + 7x^3 +0x^2 + x - 11 / x+3

Step 8: If you have a number left over, this is the remainder. Write it as the denominator of the divisor.

__-3| 3 7 0 1 -11

______________-9____6___-18___51

3 -2 6 -17 |40|

3x^3 - 2x^2 + 6x -17 +| 40 |

|x + 3|

SOLVE: 3x^4 + 7x^3 + x - 11 / x+3

Turned into...

-SOLVE: 3x^4 + 7x^3 +0x^2 + x - 11 / x+3

Using synthetic division, we found that the quotient is....

3x^3 - 2x^2 + 6x -17 + 40

x + 3

SOLVE: x^4 + 3x^3 - 6x^2 - 6x + 8 / x+4

Step 1:

x^4 + 3x^3 - 6x^2 - 6x + 8

Looks good!

SOLVE: x^4 + 3x^3 - 6x^2 - 6x + 8 / x+4

Step 2:

x^4 + 3x^3 - 6x^2 - 6x + 8

__-4| 1 3 -6 -6 8