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Synthetic DivisionBy:  Alyssa Barnett
     Synthetic DivisionSynthetic 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
         Steps to Solving1. Write the coefficients down in order2. Draw a box and line, then switch the sign of what you are dividing by3. Drop the first number down. That number stays the same4.  From then on out, multiply by the divisor, add numbers, then repeat for the rest of the problem
             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_______________________
         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.
        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 themNow, multiply the -4 and -5, then add
/-4/       3         7        -9      12_______-12___ 20__ -44___3        -5        11      -32The 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.
             RemainderThe 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.
      Synthetic DivisionThe finishing problem will look like this…3x^2 – 5x + 11 + -32/x + 4The 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.

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Synthetic division

  • 1. Synthetic DivisionBy: Alyssa Barnett
  • 2. Synthetic DivisionSynthetic 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 Solving1. Write the coefficients down in order2. Draw a box and line, then switch the sign of what you are dividing by3. Drop the first number down. That number stays the same4. 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 themNow, multiply the -4 and -5, then add
  • 7. /-4/ 3 7 -9 12_______-12___ 20__ -44___3 -5 11 -32The 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. RemainderThe 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 DivisionThe finishing problem will look like this…3x^2 – 5x + 11 + -32/x + 4The 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.