Simple algebraic division

Edexcel International GCSE Further Pure Mathematics· 3A· 15 min read

1. Key Definitions & Core Division Principle★★☆☆☆⏱ 3 min

📘 Definition

Algebraic Division Identity

For any polynomial dividend $P (x)$ and linear divisor $D (x)$ : $P (x) = D (x) \times Q (x) + R$ , where $Q (x)$ is the quotient polynomial and $R$ is the constant remainder.

Example:

Dividing $3 x^{3} + 2 x^{2} - x + 4$ by $(x - 2)$ follows the identity: $3 x^{3} + 2 x^{2} - x + 4 = (x - 2) Q (x) + R$

When dividing a polynomial of degree $n$ by a linear (degree 1) divisor, the quotient will always be a polynomial of degree $n - 1$ , and the remainder will be a constant. This rule lets you check the validity of your result before you finish calculations.

2. Method 1: Algebraic Long Division★★★☆☆⏱ 4 min

✓ Calculator OK

Algebraic long division works identically to numerical long division, except you operate on terms of $x$ instead of digits. Follow the cycle: divide leading terms, multiply divisor by the result, subtract from the current dividend segment, bring down the next term, repeat.

📐 Worked Example

Divide $2 x^{3} + 5 x^{2} - x - 6$ by $(x + 2)$ . State the quotient and remainder.

1
1. Divide the leading term of the dividend by the leading term of the divisor: $2 x^{3} \div x = 2 x^{2}$ . Write this as the first term of the quotient.
2
1. Multiply the entire divisor by $2 x^{2}$ : $2 x^{2} (x + 2) = 2 x^{3} + 4 x^{2}$ . Subtract this from the first two terms of the dividend: $(2 x^{3} + 5 x^{2}) - (2 x^{3} + 4 x^{2}) = x^{2}$ .
3
1. Bring down the next term $- x$ to get $x^{2} - x$ . Divide leading terms: $x^{2} \div x = x$ . Add $x$ to the quotient.
4
1. Multiply the divisor by $x$ : $x (x + 2) = x^{2} + 2 x$ . Subtract: $(x^{2} - x) - (x^{2} + 2 x) = - 3 x$ .
5
1. Bring down the final term $- 6$ to get $- 3 x - 6$ . Divide leading terms: $- 3 x \div x = - 3$ . Add $- 3$ to the quotient.
6
1. Multiply the divisor by $- 3$ : $- 3 (x + 2) = - 3 x - 6$ . Subtract: $(- 3 x - 6) - (- 3 x - 6) = 0$ .
7
Final result: Quotient = $2 x^{2} + x - 3$ , Remainder = 0

Exam tip:

Show all long division steps in your exam to earn method marks even if you make a small arithmetic error.

3. Method 2: Equating Coefficients★★★☆☆⏱ 4 min

✓ Calculator OK

This method uses the algebraic division identity directly. You first write the general form of the quotient based on its expected degree, expand the right-hand side of the identity, then match coefficients of like powers of $x$ to solve for unknown constants in the quotient and remainder.

📐 Worked Example

Divide $3 x^{3} - 4 x^{2} + 2 x - 1$ by $(2 x - 1)$ . State the quotient and remainder.

1
1. The dividend is degree 3, so the quotient is quadratic: let $Q (x) = A x^{2} + B x + C$ , remainder = $R$ (constant).
2
1. Write the identity: $3 x^{3} - 4 x^{2} + 2 x - 1 = (2 x - 1) (A x^{2} + B x + C) + R$
3
1. Expand the right-hand side: $2 A x^{3} + (2 B - A) x^{2} + (2 C - B) x + (R - C)$
4
1. Equate coefficients of like terms:
5
$x^3: 2A = 3 \implies A = \frac{3}{2}$
6
$x^2: 2B - A = -4 \implies 2B - \frac{3}{2} = -4 \implies B = -\frac{5}{4}$
7
$x: 2C - B = 2 \implies 2C + \frac{5}{4} = 2 \implies C = \frac{3}{8}$
8
$constant: R - C = -1 \implies R = -1 + \frac{3}{8} = -\frac{5}{8}$
9
Final result: Quotient = $\frac{3}{2} x^{2} - \frac{5}{4} x + \frac{3}{8}$ , Remainder = $- \frac{5}{8}$

Exam tip:

This method is often faster for linear divisors and reduces the risk of subtraction errors common in long division.

4. Exam-Style Practice★★★★☆⏱ 4 min

✓ Quick check

When dividing $4 x^{4} + 2 x^{3} - x + 5$ by $(3 x + 2)$ , what is the degree of the quotient?
- A) 3
- B) 4
- C) 2
- D) 1
Reveal answer
A) 3 —
The quotient degree is always equal to the dividend degree minus 1 for linear divisors: 4 - 1 = 3.
If $(x - 2) (x^{2} + 3 x + 1) + R = x^{3} + x^{2} - 5 x + 2$ , what is the value of $R$ ?
Reveal answer
4 —
Expand the left-hand side: $(x - 2) (x^{2} + 3 x + 1) = x^{3} + x^{2} - 5 x - 2$ . Rearranging gives $R = (x^{3} + x^{2} - 5 x + 2) - (x^{3} + x^{2} - 5 x - 2) = 4$ .

📐 Worked Example

Divide $x^{4} - 3 x^{2} + x + 7$ by $(x - 3)$ . State the quotient and remainder.

1
1. First rewrite the dividend with all powers of $x$ present, inserting 0 coefficients for missing terms: $x^{4} + 0 x^{3} - 3 x^{2} + x + 7$ .
2
1. Use the long division method, aligning like terms correctly throughout the process.
3
1. After completing all division steps, the resulting quotient is $x^{3} + 3 x^{2} + 6 x + 19$ , and the remainder is 64.
4
1. Verify using the identity: $(x - 3) (x^{3} + 3 x^{2} + 6 x + 19) + 64 = x^{4} - 3 x^{2} + x + 7$ , which matches the original dividend.

5. Common Pitfalls

Wrong move:

Forgetting to include missing terms (e.g., writing $x^{3} + 2 x + 1$ instead of $x^{3} + 0 x^{2} + 2 x + 1$ ) when dividing.

Why:

Missing terms lead to incorrect alignment of like terms during calculation, resulting in wrong quotient coefficients.

Correct move:

Always rewrite the dividend with all powers of $x$ present, inserting 0 coefficients for any missing terms before starting division.

Wrong move:

Making sign errors when subtracting products during long division.

Why:

Sign errors are the most common mistake in algebraic long division, and often lead to lost method and answer marks even if your process is otherwise correct.

Correct move:

Write brackets around the product you are subtracting, then distribute the negative sign before combining terms to avoid sign errors.

Wrong move:

Using an incorrect degree for the quotient when using the equating coefficients method.

Why:

An incorrectly sized quotient will have missing or extra terms, making it impossible to match coefficients correctly.

Correct move:

Always calculate the expected quotient degree as (dividend degree - 1) before writing the general form of the quotient.

Wrong move:

Omitting the remainder when presenting your final answer.

Why:

Exam questions explicitly ask for both quotient and remainder, so omitting either will cost you answer marks even if your quotient is correct.

Correct move:

Always present your final answer in the form $P (x) = D (x) Q (x) + R$ , or clearly label and state both the quotient and remainder separately.

6. Quick Reference Cheatsheet

Divisor Type	Quotient Degree (for degree n dividend)	General Identity	Approved Methods
$(x \pm a)$	$n - 1$	$P (x) = (x \pm a) Q (x) + R$	Long division, Equating coefficients
$(a x \pm b)$	$n - 1$	$P (x) = (a x \pm b) Q (x) + R$	Long division, Equating coefficients

Going deeper

syllabusEdexcel 4PM1 Specification

What's Next

Now that you have mastered simple algebraic division, you are ready to move on to the factor and remainder theorems, which build directly on the quotient and remainder rules you learned here. This topic forms the foundation for solving higher-degree polynomial equations and simplifying algebraic expressions that appear across the Edexcel IGCSE Further Pure Maths syllabus, including in coordinate geometry and calculus questions. Practice both division methods regularly to build speed and accuracy for your exam, as they are frequently tested in both short and long answer questions.