Dividing an algebraic expression by a monomial means dividing every term in the expression by that single term separately. Each division reduces the power and cancels numerical common factors, resulting in a simpler expression. This skill is fundamental for rearranging formulae and simplifying algebraic fractions.
What is a monomial?
A monomial is a single algebraic term — a number, a variable, or a product of numbers and variables with no addition or subtraction. Examples: 3, x, 4x, 2x², −5xy.
An expression such as 6x² + 4x is not a monomial — it is a binomial (two terms). Dividing a binomial (or polynomial) by a monomial means applying the division to each term separately.
How do you divide a binomial by a monomial?
The rule: split the expression so that each term in the numerator has its own copy of the denominator, then simplify each fraction separately.
$$\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}$$
Worked example 1: Simplify (6x² + 4x) ÷ 2x
- Write as a fraction: $\frac{6x^2 + 4x}{2x}$
- Split: $\frac{6x^2}{2x} + \frac{4x}{2x}$
- First term: $\frac{6}{2} \times \frac{x^2}{x} = 3x$
- Second term: $\frac{4}{2} \times \frac{x}{x} = 2$
- Answer: 3x + 2
Always check: multiply the answer by the divisor: (3x + 2) × 2x = 6x² + 4x ✓
How do you cancel powers of variables?
When dividing variables with indices, subtract the powers (this is the index law for division):
$$x^m \div x^n = x^{m-n}$$
Worked example 2: Simplify (12x³ − 8x²) ÷ 4x
| Term | Numerator | Division | Simplified |
|---|---|---|---|
| First | 12x³ ÷ 4x | 12÷4 = 3, x³÷x = x² | 3x² |
| Second | −8x² ÷ 4x | −8÷4 = −2, x²÷x = x | −2x |
Answer: 3x² − 2x
How do you divide by a monomial with two variables?
Treat each variable independently.
Worked example 3: Simplify (10x²y + 15xy²) ÷ 5xy
- Write as a fraction: $\frac{10x^2y + 15xy^2}{5xy}$
- Split: $\frac{10x^2y}{5xy} + \frac{15xy^2}{5xy}$
- First term: $\frac{10}{5} \cdot \frac{x^2}{x} \cdot \frac{y}{y} = 2x$
- Second term: $\frac{15}{5} \cdot \frac{x}{x} \cdot \frac{y^2}{y} = 3y$
- Answer: 2x + 3y
What if the divisor does not cancel completely?
Sometimes the numerical factor does not divide evenly — leave it as a fraction.
Worked example 4: Simplify (9x² + 6x) ÷ 3x²
- Split: $\frac{9x^2}{3x^2} + \frac{6x}{3x^2}$
- First term: $\frac{9}{3} \cdot \frac{x^2}{x^2} = 3$
- Second term: $\frac{6}{3} \cdot \frac{x}{x^2} = 2 \cdot x^{-1} = \frac{2}{x}$
- Answer: 3 + 2/x
Note: the second term gives a negative index (x⁻¹ = 1/x). At KS3 it is usually cleaner to write it as 2/x rather than 2x⁻¹ unless told otherwise.
What mistakes should you avoid?
- Dividing only the first term. Every term in the numerator must be divided. A common error is to write (6x² + 4x) ÷ 2x = 3x + 4x = 7x, leaving the second term undivided.
- Subtracting powers incorrectly. x³ ÷ x = x³⁻¹ = x², not x³. Subtract — do not multiply or divide — the indices.
- Forgetting to halve the coefficient. When dividing 8x by 4, the numerical result is 2, not 4. Always divide the numbers and the variables separately.
- Changing the sign. Dividing a negative term by a positive divisor gives a negative result; dividing two negatives gives a positive. Apply the sign rules for division, not just for multiplication.
Frequently asked questions
Why do we split the fraction into separate terms?
Because addition and subtraction distribute over division: (a + b)/c = a/c + b/c. This is the same reason you can distribute multiplication over a bracket: 3(a + b) = 3a + 3b. Division by a monomial is essentially "factoring out" the divisor. If the divisor had been a binomial (two terms), this splitting trick would not work — that requires polynomial long division, which is a GCSE Higher topic.
How do I check my answer?
Multiply your simplified answer by the monomial you divided by. If you get back the original expression, the answer is correct. This is the inverse operation: (answer) × (divisor) = original. For Worked example 1: (3x + 2) × 2x = 6x² + 4x ✓
What if the divisor is just a number with no variable?
Divide each coefficient by that number, leaving the variable parts unchanged. For example, (12x² − 8x + 4) ÷ 4 = 3x² − 2x + 1. This is equivalent to "factorising out" the common factor.
Where does this skill appear later in maths?
Dividing by a monomial is used when: (1) rearranging formulae (isolating a variable that appears as part of a product), (2) simplifying algebraic fractions at GCSE, (3) integrating polynomials at A-level (dividing each term of the integrand), and (4) simplifying rational expressions in Further Maths. Mastering it at KS3 means these later topics feel natural rather than new.
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