Algebraic long division divides a polynomial by a linear expression using the same method as long division with numbers: divide the leading terms, multiply back, subtract, and bring down the next term. It gives the quotient and any remainder, handles missing powers of x with zero placeholders, and, once you know one factor of a cubic, lets you find the remaining quadratic and solve completely.
Algebraic division
- f(x) = (divisor) × (quotient) + remainder
- Dividing by a linear expression leaves a constant remainder
- Insert 0 for any missing power, such as x⁴ + 0x³ + 0x² + 0x − 1
- Remainder when dividing by (x − a) equals f(a)
How does polynomial long division work?
Set it out like long division with numbers, keeping each power of x in its own column. Repeat four steps until no terms are left to bring down: divide the leading term by the leading term of the divisor, multiply the divisor by that result, subtract, and bring down the next term.
The first term of the quotient is the leading term of the dividend divided by the leading term of the divisor, such as x³ ÷ x = x².
Division with a remainder
If the division does not come out exactly, the final subtraction leaves a remainder. Write the result as f(x) = (divisor) × (quotient) + remainder. For a linear divisor, the remainder is a constant. You can check it using the remainder theorem: dividing by (x − a) leaves remainder f(a).
Missing terms and higher powers
If a power of x is missing from the dividend, write it in with a coefficient of zero so that the columns stay lined up. For example, x⁴ − 1 is written as x⁴ + 0x³ + 0x² + 0x − 1. A quartic divided by a linear expression gives a cubic quotient.
Dividing by ax − b
The method is the same with a non-monic divisor such as 2x + 1. Take care with the first step of each round: 2x³ ÷ 2x = x². Most errors happen when multiplying the divisor back through, so check each product.
Factorising and solving using division
Once you know one factor, (x − a), divide by it to get the remaining quadratic, then factorise that quadratic and set each factor to zero. This solves a cubic completely. If the quadratic does not factorise, use the formula and give the roots exactly.
Worked examples
Example 1. Divide x³ − 4x² + x + 6 by (x − 3).
- x³ ÷ x = x². Multiply: x³ − 3x². Subtract: −x², and bring down + x.
- −x² ÷ x = −x. Multiply: −x² + 3x. Subtract: −2x, and bring down + 6.
- −2x ÷ x = −2. Multiply: −2x + 6. Subtract: 0.
Answer: Quotient x² − x − 2, remainder 0.
Marks: M1 for the first term x², M1 for subtracting and bringing down, A1 for the quotient.
Example 2. Divide x³ + 2x² − 3x + 5 by (x − 1), and write the result in quotient-remainder form.
- Long division gives quotient x² + 3x and remainder 5.
- So x³ + 2x² − 3x + 5 = (x − 1)(x² + 3x) + 5.
- Check: f(1) = 1 + 2 − 3 + 5 = 5.
Answer: (x − 1)(x² + 3x) + 5.
Marks: M1 for the quotient, A1 for the remainder, B1 for the check.
Example 3. Divide x⁴ − 1 by (x − 1).
- Write the dividend with placeholders: x⁴ + 0x³ + 0x² + 0x − 1.
- Divide column by column; each step brings down a zero.
Answer: x³ + x² + x + 1, with remainder 0.
Marks: M1 for the placeholders, A1 for the quotient.
Example 4. Divide 2x³ + 3x² − 11x − 6 by (2x + 1).
- 2x³ ÷ 2x = x². Multiply: 2x³ + x². Subtract: 2x².
- 2x² ÷ 2x = x. Multiply: 2x² + x. Subtract: −12x, and bring down −6.
- −12x ÷ 2x = −6. Multiply: −12x − 6. Subtract: 0.
Answer: x² + x − 6.
Marks: M1 for 2x³ ÷ 2x = x², M1 for each subtraction, A1 for the quotient.
Example 5. Given that (x − 2) is a factor of x³ − x² − 4x + 4, solve x³ − x² − 4x + 4 = 0.
- Divide by (x − 2) to get x² + x − 2.
- Factorise: x² + x − 2 = (x + 2)(x − 1).
- So (x − 2)(x + 2)(x − 1) = 0.
Answer: x = 2, x = −2 or x = 1.
Marks: M1 for the division, M1 for factorising, A1 for all three roots.
Common mistakes
- Leaving out missing powers. Use 0 placeholders so each column holds one power of x.
- Sign errors when subtracting. Subtract every term of the product line, not just the first.
- Dividing by the wrong leading term. With 2x + 1, divide by 2x, not by x.
- Stopping too early. Keep going until there is nothing left to bring down.
- Not solving the quadratic. After dividing, factorise or solve the quadratic to find every root.
How algebraic division is examined
Algebraic division is part of algebraic methods, the opening chapter of Pure 2 (WMA12) in Edexcel International A-Level. In Edexcel A-Level (9MA0) it is first-year Pure. Typical tasks:
- divide a cubic by a linear factor and state the quotient and remainder
- use a given factor to divide and then solve a cubic fully
- give irrational roots in exact surd form
- divide by a non-monic linear expression
These questions, with mark schemes, are in the Pure 2 algebraic methods exam questions. The algebraic methods worksheet and the algebra revision engine give further practice.
Division pairs naturally with the factor and remainder theorem, and it reappears in improper algebraic fractions.
Frequently asked questions
How do you divide a polynomial by (x − a)?
Use long division: divide the leading terms, multiply back, subtract, bring down the next term, and repeat until no terms are left.
What is the remainder when dividing by a linear expression?
A constant. By the remainder theorem, dividing f(x) by (x − a) leaves remainder f(a).
Why do I need zero placeholders?
They keep each power of x in the right column. Without them, terms get subtracted from the wrong powers.