Malone Global University Logo Malone Global University

Malone Algebra - Unit 7

Lesson 43: Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − c. It is faster and simpler than long division for these cases.

Steps for Synthetic Division

  1. Write down the coefficients of the dividend polynomial in order.
  2. Write c (from x − c) to the left.
  3. Bring down the first coefficient.
  4. Multiply it by c and write the result under the next coefficient.
  5. Add the column.
  6. Repeat multiply and add until all coefficients are processed.
  7. The last number is the remainder; the others are coefficients of the quotient polynomial.

Example

Divide f(x) = 2x³ − 3x² + 4x − 5 by x − 2

  1. Coefficients: 2, −3, 4, −5
  2. c = 2 (from x − 2)
  3. Set up synthetic division:
    4.           2 | 2  -3  4  -5
             |____
  4. Bring down 2 → multiply 2×2=4 → add −3+4=1
  5. Multiply 1×2=2 → add 4+2=6
  6. Multiply 6×2=12 → add −5+12=7
  7. Quotient: 2x² + x + 6, Remainder: 7
  8. Result: f(x)/(x−2) = 2x² + x + 6 + 7/(x−2)

Practice Problems

  1. Divide x³ + 3x² − 4 by x + 2
  2. Divide 3x³ − 5x² + 2x − 7 by x − 1
  3. Divide 2x⁴ − 3x³ + x² − 4 by x + 2
  4. Divide x³ − 6x² + 11x − 6 by x − 1
  5. Divide 4x³ + x² − 5x + 2 by x − 2