Synthetic division is a method of polynomial long division that is more efficient by omitting variables and exponents. It involves writing the coefficients of a polynomial in order and performing division on the list of coefficients, switching the sign of the divisor. You bring down the first coefficient and multiply and add throughout the process. Any remainder at the end based on the last numbers not cancelling out is added above the original divisor equation.

1. Synthetic Division By: Alyssa Barnett

2. Synthetic Division Synthetic division is a method of long division, but only using less writing and not as much solving. a method of dividing polynomials in which you leave out all variables and exponents and perform division on the list of coefficients. You also switch the sign of the divisor so that you can add throughout the process

3. Steps to Solving 1. Write the coefficients down in order 2. Draw a box and line, then switch the sign of what you are dividing by 3. Drop the first number down. That number stays the same 4. From then on out, multiply by the divisor, add numbers, then repeat for the rest of the problem

4. Examples (3x^3 + 7x^2 – 9x + 12) / (x + 4) In this problem you have 4 numbers to bring down. So start off by drawing a line then bringing down only the numbers. 3 7 -9 12 _______________________

5. Example cont… 3 7 -9 12 _______________________ In a little box near the three write the number that is the divisor. Which is the four. But, in synthetic you switch the sign. This will make the divisor now -4. Always in a synthetic equation you bring down the very first coefficient. The three will come down and then you start multiplying and adding.

6. Example cont… /-4/ 3 7 -9 12 _______________________ 3 Start multiplying and adding. -4 * 3 = -12 ; You now put the -12 under the 7 because you are going to add those numbers. -12 + 7= -5; You write the -5 under the line where you added them Now, multiply the -4 and -5, then add

7. /-4/ 3 7 -9 12 _______-12___ 20__ -44___ 3 -5 11 -32 The problem is finished except you have a remainder. The way to know if you have a remainder is if the very last numbers you add up together don’t cancel each other out.

8. Remainder The remainder in Synthetic Division problems is always based on the very last numbers you add up together. So, for the example we did, there was a remainder of -32. Adding 12 and -44 didn’t cancel out or equal zero. When writing the finishing problem, your numbers left under the line are what you will use. 3, -5, 11, and -32. Whatever number of (x) you used in the original problem, you use one less in the finishing problem. So, when the original problem starts off with 3x^3, you will end up with 3x^2 and keep going down until you have no more x’s.

9. Synthetic Division The finishing problem will look like this… 3x^2 – 5x + 11 + -32/x + 4 The remainder of the problem is added in the equation. The remainder on top and the original divisor equation, which was (x + 4), goes on the bottom.