Polynomial Long Division

Polynomial long division works the same way as number long division. You divide, multiply, subtract, and bring down, then repeat until there is nothing left to bring down.

Steps:

  1. Write the dividend and divisor in descending order of degree. Use zero coefficients for any missing terms.
  2. Divide the leading term of the dividend by the leading term of the divisor. Write the result above the division bar.
  3. Multiply that result by the entire divisor and write the product underneath, aligning like terms.
  4. Subtract and bring down the next term.
  5. Repeat until the degree of the remainder is lower than the degree of the divisor.

Long Division Example

Divide x³ - 7x - 6 by (x - 3). Since x² is missing, treat the dividend as x³ + 0x² - 7x - 6.

Dividend: x³ + 0x² - 7x - 6
Divisor: (x - 3)

Step 1: x³ / x = x². Multiply: x²(x - 3) = x³ - 3x².
Subtract: (x³ + 0x²) - (x³ - 3x²) = 3x².
Bring down: 3x² - 7x

Step 2: 3x² / x = 3x. Multiply: 3x(x - 3) = 3x² - 9x.
Subtract: (3x² - 7x) - (3x² - 9x) = 2x.
Bring down: 2x - 6

Step 3: 2x / x = 2. Multiply: 2(x - 3) = 2x - 6.
Subtract: (2x - 6) - (2x - 6) = 0.

Result: x² + 3x + 2, remainder 0

A remainder of 0 means (x - 3) divides evenly. So x³ - 7x - 6 = (x - 3)(x² + 3x + 2), and that quadratic factors further as (x + 1)(x + 2).

Synthetic Division

Synthetic division is a condensed version that only works when dividing by a linear binomial of the form (x - a). You work only with the coefficients instead of writing out all the variable expressions.

Steps:

  1. Write the value of a (from x - a) in a box on the left.
  2. Write the coefficients of the dividend in a row to the right. Use 0 for any missing terms.
  3. Bring down the first coefficient.
  4. Multiply it by a, write the result under the next coefficient, then add.
  5. Repeat until done. The last number is the remainder and the rest are coefficients of the quotient.

Synthetic Division Example

Divide 2x³ - 3x² - 11x + 6 by (x - 3), so a = 3.

Coefficients of dividend: 2, -3, -11, 6
Value of a: 3

Bring down 2.
3 * 2 = 6. Add to -3: gives 3.
3 * 3 = 9. Add to -11: gives -2.
3 * (-2) = -6. Add to 6: gives 0.

Quotient coefficients: 2, 3, -2 | Remainder: 0

Result: 2x² + 3x - 2, remainder 0

The remainder of 0 confirms that (x - 3) is a factor. The quotient 2x² + 3x - 2 factors as (2x - 1)(x + 2).

Which Method to Use

Use synthetic division when the divisor is a linear binomial (x - a) with a leading coefficient of 1. It is faster and easier to get right.

Use long division when the divisor has a leading coefficient other than 1, when the divisor is quadratic or higher degree, or when you just want to see every step clearly.

Both methods give the same answer. Synthetic division is just a shortcut notation, not a different process.