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Synthetic Division

Long division is useful with the remainder and factor theorems, but long division can be time consuming. To divide a polynomial by a binomial and compute the remainder, we can also use synthetic division. We can only divide by a binomial whose leading coefficient is 1--thus, we must factor the leading coefficient out of the binomial and divide by the leading coefficient separately. Also, the binomial must have degree 1; we cannot use synthetic division to divide by a binomial like x2 + 1. Here are the steps for dividing a polynomial by a binomial using synthetic division:

  1. Write the polynomial in descending order, adding "zero terms" if an exponent term is skipped.
  2. If the polynomial does not have a leading coefficient of 1, write the binomial as b(x - a) and divide the polynomial by b. Otherwise, leave the binomial as x - a.
  3. Write the value of a, and write all the coefficients of the polynomial in a horizontal line to the left of a.
  4. Draw a line below the coefficients, leaving room above the line.
  5. Bring the first coefficient below the line.
  6. Multiply the number below the line by a and write the result above the line below the next coefficient.
  7. Subtract the result from the coefficient above it.
  8. Repeat steps 6 and 7 until all the coefficients have been used.
  9. If the polynomial has n terms, the first n - 1 numbers below the line are the coefficients of the resulting polynomial, and the last number is the remainder.


Example: What is the result when 4x4 -6x3 -12x2 - 10x + 2 is divided by x - 3? What is the remainder?

Figure %: Synthetic Division
The result is 4x3 +6x2 + 6x + 8, and the remainder is 26.

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