How to Divide Polynomials Using Synthetic Division

Step-by-Step Guide

1. 1

   Step 1: Write down the problem.

   For this example, you will be dividing x3 + 2x2
   - 4x + 8 by x +
   2.
   
   Write the first polynomial equation, the dividend, in the numerator and write the second equation, the divisor, in the denominator.
2. 2

   Step 2: Reverse the sign of the constant in the divisor.

   The constant in the divisor, x + 2, is positive 2, so reversing the sign of the constant would give you
   -2. , The upside-down division symbol will look a bit like a backwards "L." Place the term
   -2 to the left of this symbol. , Write the terms from left to right, just as they appear.
   
   It should look like this:
   -2| 1 2
   -4
   8. , Bring down the first coefficient, 1, below itself.
   
   It should look like this:
   -2| 1  2  -4  8     ↓     1 , Simply multiply 1 by
   -2 to get
   -2 and write this product under the second term,
   2.
   
   Here's how it would look:
   -2| 1  2  -4  8         -2     1 , Now take the second coefficient, 2, and add it to
   -2.
   
   The result is
   0.
   
   Write this result below the two numbers, just as you would in long division.
   
   Here's how it would look:
   -2| 1  2  -4  8         -2     1   0 , Now, take the sum, 0, and multiply it by the divisor,
   -2.
   
   The result is
   0.
   
   Place this number below 4, the third coefficient.
   
   It should look like this:
   -2| 1  2  -4  8         -2  0      1    , Add 0 and
   -4 to get
   -4 and write this answer below the
   0.
   
   Here's how it would look:
   -2| 1  2  -4  8         -2   0      1   0   -4 , Now, multiply
   -4 by
   -2 to get 8, write this answer under the fourth coefficient, 8, and add this answer to the fourth coefficient. 8 + 8 = 16, so this is your remainder.
   
   Write this number below the product.
   
   Here's how it would look:
   -2| 1  2  -4  8         -2   0   8     1   0   -4   |16 , In this case, the first sum, 1, is placed next to an x to the second power (one less than three).
   
   The second sum, 0, is placed next to an x, but the result is zero, so you can remove this term.
   
   And the third coefficient,
   -4, becomes a constant, a number without a variable, since the original variable was x.
   
   You can write an R next to the 16, because that is the remainder.
   
   Here's how it would look:
   -2| 1  2  -4  8         -2   0   8     1   0   -4   |16     x2   + 0x   
   - 4    R 16x2
   - 4 R16 , The final answer is the new polynomial, x2
   - 4, plus the remainder, 16, over the original divisor, x +
   2.
   
   Here's how it would look: x2
   - 4 +16/(x +2).
3. 3

   Step 3: Place this number outside the upside-down division symbol.

4. 4

   Step 4: Write all of the coefficients of the dividend inside the division symbol.

5. 5

   Step 5: Bring down the first coefficient.

6. 6

   Step 6: Multiply the first coefficient by the divisor and place it under the second coefficient.

7. 7

   Step 7: Add the second coefficient and the product and write the answer below the product.

8. 8

   Step 8: Multiply this sum by the divisor and place the result under the third coefficient.

9. 9

   Step 9: Add the product and the third coefficient and write the result under the product.

10. 10

    Step 10: Multiply this number by the divisor

11. 11

    Step 11: write it under the last coefficient

12. 12

    Step 12: and add it to the coefficient.

13. 13

    Step 13: Place each of the new coefficients next to a variable of one less power than their original corresponding variables.

14. 14

    Step 14: Write the final answer.

Detailed Guide

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