Polynomial Long Division, Step by Step

If you've done long division with plain numbers, you already know the rhythm: divide, multiply, subtract, bring down the next term, and repeat until nothing's left to bring down.

The steps, laid out:

  1. Arrange the dividend and divisor in descending order by degree, inserting a coefficient of 0 for any missing power of x.
  2. Divide the leading term of the dividend by the leading term of the divisor, and write the result above the division bar.
  3. Multiply that result by the entire divisor, then write the product underneath, lining up matching terms.
  4. Subtract, then bring down the next term from the dividend.
  5. Repeat until what remains has a lower degree than the divisor.

Worked Example: Long Division

Divide x³ - 7x - 6 by (x - 3). Since there's no x² term written, insert 0x² to keep the columns lined up: 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 in evenly, so x³ - 7x - 6 = (x - 3)(x² + 3x + 2). That leftover quadratic factors further into (x + 1)(x + 2).

Synthetic Division: The Shortcut

Synthetic division is a compressed version of the same process, limited to cases where you're dividing by a linear binomial like (x - a). Rather than writing out each term with its variable, you work only with the coefficients.

Here's the routine:

  1. Take the value a from (x - a) and set it off to the left in a small box.
  2. List the dividend's coefficients in a row, filling in 0 wherever a term is missing.
  3. Drop the first coefficient straight down.
  4. Multiply that number by a, add the result to the next coefficient, and write the sum below.
  5. Keep multiplying and adding across the row. The last number is the remainder, and everything before it forms the quotient's coefficients.

Worked Example: Synthetic Division

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

That zero remainder confirms (x - 3) is a genuine factor here. The quotient 2x² + 3x - 2 breaks down further into (2x - 1)(x + 2).

Which Method Should You Use?

Reach for synthetic division when the divisor is a simple linear binomial (x - a) with a leading coefficient of 1. It's faster and leaves less room for arithmetic slip-ups.

Stick with long division when the divisor's leading coefficient isn't 1, when the divisor is quadratic or higher, or when you want every intermediate step spelled out.

Either way you arrive at the same quotient and remainder. Synthetic division isn't a separate technique under the hood - it's just long division written more efficiently for linear divisors.