Algebraic division

The layout

Divide the leading term, multiply back, subtract, bring down the next term, and repeat.

For (x³ + 2x² - 5x - 6) ÷ (x + 1):

• x³ ÷ x = x²; subtract x²(x + 1) to leave x² - 5x
• x² ÷ x = x; subtract x(x + 1) to leave -6x - 6
• -6x ÷ x = -6; subtract -6(x + 1) to leave 0

Missing terms

Write any missing power with a 0 coefficient so the columns line up: x³ - 7x - 6 becomes x³ + 0x² - 7x - 6.

What it means

f(x) = (divisor)(quotient) + remainder. If the remainder is 0, the divisor is a factor and you can factorise f(x) further.

Worked example

Divide x³ + 2x² - 5x - 6 by (x + 1) and factorise fully.

1. Quotient x² + x - 6, remainder 0
2. So x³ + 2x² - 5x - 6 = (x + 1)(x² + x - 6)
3. x² + x - 6 = (x + 3)(x - 2)

Answer: (x + 1)(x + 3)(x - 2)

Check yourself

1. Find the quotient when x³ - 4x² + x + 6 is divided by (x - 2).
   Show answer

   x² - 2x - 3. x³ ÷ x = x², then -2x² ÷ x = -2x, then -3x ÷ x = -3. The remainder is 0.

2. Find the remainder when 2x³ + x² - 4 is divided by (x - 2).
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   16. Dividing gives quotient 2x² + 5x + 10 and remainder 16. (Check: 2(8) + 4 - 4 = 16.)

3. Find the quotient when 2x³ + 3x² - 8x + 3 is divided by (2x - 1).
   Show answer

   x² + 2x - 3. 2x³ ÷ 2x = x²; then 4x² ÷ 2x = 2x; then -6x ÷ 2x = -3. The remainder is 0.

4. What is the quotient when x³ - 7x - 6 is divided by (x + 1)?
   • x² + x - 6
   • x² - 7x - 6
   • x² - x - 6
   • x² - x - 8
   Show answer

   x² - x - 6. Write x³ + 0x² - 7x - 6. Then x², then -x² ÷ x = -x, then -6x ÷ x = -6.

5. Why do you write x³ - 7x - 6 as x³ + 0x² - 7x - 6 before dividing?
   • So that every power has its own column
   • Because x² is a factor
   • To make the remainder zero
   • Because the quotient must contain x²
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   So that every power has its own column. The 0x² keeps the like terms lined up during subtraction.

6. x³ + 2x² - 5x - 6 = (x + 1)(x² + ax + b). Find b.
   Show answer

   -6. The quotient is x² + x - 6, so b = -6.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 2 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.