How to Perform Synthetic Division: A Step-by-Step Tutorial

Synthetic division is a fast, shorthand method for dividing a polynomial by a linear divisor of the form (x - c). It's much quicker than long division, and once you learn the steps, you can do it in minutes. This guide will walk you through the process by hand, with clear examples and tips to avoid common mistakes. If you just want the answer, use our Synthetic Division Calculator to see the result instantly.

Before we start, you may want to review What is Synthetic Division? for a conceptual overview, or check the Synthetic Division Formula to understand the algebra behind it.

You'll Need:

• A polynomial written in descending order of powers (e.g., 2x³ + 5x² - 3x + 7)
• A linear divisor of the form (x - c) — note that (x + 3) becomes (x - (-3)), so c = -3
• Paper and pencil
• Basic arithmetic skills (addition and multiplication)

Step-by-Step Guide

Follow these steps to perform synthetic division manually.

1. Identify the coefficients and c. Write the polynomial in descending order. If any term is missing, use 0 as its coefficient. The divisor should be in the form x - c; if it's x + a, then c = -a.
2. Set up the synthetic division table. Draw a small L-shaped box. Write the coefficients of the polynomial in a row inside the box (from highest degree to constant). Write the value of c outside the box to the left.
3. Bring down the first coefficient. Draw a horizontal line below the row. Copy the first coefficient directly below the line.
4. Multiply and add. Multiply the number below the line (just brought down or added) by c. Write the product in the next column, above the line. Then add that product to the coefficient above it. Write the sum below the line.
5. Repeat for all columns. Continue the multiply-add process across all coefficients. The last column gives you the remainder.
6. Interpret the result. The numbers below the line (except the last) are the coefficients of the quotient polynomial, starting from one degree less than the original. The final number is the remainder. The relationship is: P(x) = (x - c) × Q(x) + R.

Worked Examples

Example 1: (2x³ + 5x² - 3x + 7) ÷ (x - 2)

We have polynomial 2x³ + 5x² - 3x + 7, degree 3, and divisor (x - 2) so c = 2.

1. Coefficients: [2, 5, -3, 7].
2. Set up: Write c=2 to the left, coefficients in a row: 2, 5, -3, 7.
3. Bring down the first coefficient (2).
4. Multiply 2 × 2 = 4, write under 5, add: 5+4=9. Multiply 9 × 2 = 18, write under -3, add: -3+18=15. Multiply 15 × 2 = 30, write under 7, add: 7+30=37.
5. The bottom row: 2, 9, 15, 37. Last number is remainder R=37. The first three are quotient coefficients: 2 (x² term), 9 (x term), 15 (constant). So quotient Q(x) = 2x² + 9x + 15.

Verification: (x - 2)(2x² + 9x + 15) + 37 = 2x³ + 9x² + 15x - 4x² - 18x - 30 + 37 = 2x³ + 5x² - 3x + 7. Correct.

Example 2: (x⁴ - 16) ÷ (x + 3)

Polynomial x⁴ - 16, degree 4. Missing terms: x³, x², x. Coefficients: [1, 0, 0, 0, -16]. Divisor (x + 3) means c = -3.

1. Coefficients: 1, 0, 0, 0, -16; c = -3.
2. Bring down 1. Multiply 1 × (-3) = -3, add to 0 → -3.
3. Multiply -3 × (-3) = 9, add to 0 → 9.
4. Multiply 9 × (-3) = -27, add to 0 → -27.
5. Multiply -27 × (-3) = 81, add to -16 → 65.
6. Bottom row: 1, -3, 9, -27, 65. Remainder R=65. Quotient coefficients: 1 (x³), -3 (x²), 9 (x), -27 (constant). So Q(x) = x³ - 3x² + 9x - 27.

Verification: (x + 3)(x³ - 3x² + 9x - 27) + 65 = x⁴ - 3x³ + 9x² - 27x + 3x³ - 9x² + 27x - 81 + 65 = x⁴ - 16. Correct.

Common Pitfalls

• Missing terms: Always include zero coefficients for missing powers. For example, x³ - 1 becomes [1, 0, 0, -1].
• Sign errors: Be careful when c is negative. Double-check multiplication and addition.
• Confusing quotient and remainder: The last number is the remainder; the others are the quotient coefficients. The degree of the quotient is one less than the original polynomial's degree.
• Forgetting to write the divisor correctly: For x + 5, c = -5; not +5.

For more practice interpreting your results, visit Interpreting Synthetic Division Results.