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Synthetic Division

A free College Algebra lesson from the “Polynomial Arithmetic and Theorems” unit, with a worked example and practice problems including step-by-step solutions.

Synthetic division is a compact shortcut for dividing a polynomial by a linear factor of the form (x - c). Write c, then the coefficients of the polynomial. Bring down the first coefficient. Multiply by c, add into the next coefficient. Repeat. The last number is the remainder; the others are the quotient coefficients (one degree lower than the dividend).

What you'll learn

Why it matters: Engineering models that need quick polynomial factoring (transfer functions, control systems) and algebra software all rely on synthetic division as the basic polynomial-division algorithm.

Worked example

Problem. Divide x^3 - 7x + 6 by (x - 1) using synthetic division. Enter the remainder.

  1. Coefficients: 1, 0, -7, 6 (note the 0 for the missing x^2 term).
  2. Synthetic with c = 1: bring 1; 1*1 = 1, 0+1 = 1; 1*1 = 1, -7+1 = -6; 1*(-6) = -6, 6+(-6) = 0.
  3. Bottom row: 1, 1, -6, 0. Quotient x^2 + x - 6, remainder 0.

Answer: 0

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. Divide x^2 + 5x + 6 by (x + 2) using synthetic division. Enter the remainder.

Show solution
  1. c = -2. Coefficients 1, 5, 6.
  2. Synthetic: 1; 1*-2=-2, 5+-2=3; 3*-2=-6, 6+-6=0. Remainder 0.

Answer: 0

2. Same problem. Enter the linear coefficient of the quotient.

Show solution
  1. Quotient is x + 3 (from bottom row 1, 3 with remainder 0).

Answer: 1

3. Divide x^2 - 4 by (x - 2). Enter the remainder.

Show solution
  1. c = 2. Coefficients 1, 0, -4.
  2. 1; 2, 0+2=2; 4, -4+4=0. Remainder 0.

Answer: 0

4. Divide x^3 - 7x + 6 by (x - 1). Enter the remainder.

Show solution
  1. c = 1. Coefficients 1, 0, -7, 6.
  2. Final bottom row: 1, 1, -6, 0. Remainder 0.

Answer: 0

5. Same: divide x^3 - 7x + 6 by (x - 1). Enter the constant term of the quotient.

Show solution
  1. Quotient is x^2 + x - 6 from bottom row 1, 1, -6.
  2. Constant term is -6.

Answer: -6

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