Working with Algebraic Fractions
Manipulating algebraic expressions means simplifying them by combining terms, applying index laws, and factorising to simplify algebraic fractions (rational expressions).
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Always factorise numerators and denominators completely before attempting to cancel terms in an algebraic fraction.
- Master the laws of indices for multiplication, division, and powers of powers (e.g., x^a × x^b = x^(a+b)).
- To add or subtract algebraic fractions, you must first find a common denominator, preferably the Lowest Common Multiple (LCM) of the original denominators.
- When dividing by an algebraic fraction, multiply by its reciprocal (invert the fraction and multiply).
- Cancellation only works for common factors (multiplied terms), not for separate terms being added or subtracted.
Formulae
x^m × xn = x^(m+n) When multiplying two powers that have the same base.
x^m / xn = x^(m-n) When dividing two powers that have the same base.
(x^m)n = x^(mn) When a power is raised to another power.
x^(-n) = 1 / xn To handle negative exponents by converting them to a reciprocal with a positive exponent.
a/b - c/d = (ad - bc) / bd As a general rule for subtracting algebraic fractions by creating a common denominator.
Definitions
- Rational Expression
- A fraction where both the numerator and denominator are polynomials. For example, (x2 + 3x) / (x - 5).
- Like Terms
- Terms containing the exact same variables raised to the same powers. For example, 5xy2 and -2xy2 are like terms and can be combined.
- Factorising
- Rewriting an expression as a product of simpler expressions (its factors). For example, x2 - 4 becomes (x - 2)(x + 2).
Worked example
Simplify fully the expression: (6x2 - x - 2) / (9x2 - 4) ÷ (2x + 1) / (3x + 2)
- 1
First, change the division into a multiplication by taking the reciprocal of the second fraction:
(6x2 - x - 2) / (9x2 - 4) × (3x + 2) / (2x + 1).
- 2
Factorise all the quadratic expressions.
The numerator 6x2 - x - 2 factorises to (3x - 2)(2x + 1).
- 3
The denominator 9x2 - 4 is a difference of two squares, which factorises to (3x - 2)(3x + 2).
- 4
Substitute the factorised forms back into the expression:
[(3x - 2)(2x + 1)] / [(3x - 2)(3x + 2)] × (3x + 2) / (2x + 1).
- 5
Cancel common factors between the numerators and denominators.
(3x - 2) cancels with (3x - 2).
(2x + 1) cancels with (2x + 1).
(3x + 2) cancels with (3x + 2).
- 6
After cancelling all factors, the result is 1.
Answer: 1
Common mistakes
- ×Making sign errors, especially when subtracting a fraction. The minus sign must apply to every term in the numerator of the second fraction, not just the first term.
- ×Incorrectly cancelling individual terms instead of entire factors. You cannot cancel the x2 in (x2 + 1) / x2 because it is a term, not a factor.
- ×Incomplete factorisation. Forgetting to take out a numerical common factor first, or failing to spot a difference of two squares.
- ×Errors in basic arithmetic when expanding brackets or finding a common denominator, often due to working too quickly.
No-calculator tips
- ✓When factorising quadratics like ax2+bx+c, find the prime factors of 'ac' to systematically test pairs that sum to 'b'.
- ✓Recognise algebraic patterns instantly: difference of two squares (x2 - y2), and perfect squares ((x+y)2 or (x-y)2) to save time.
- ✓Before performing a full multiplication or expansion, check if any parts can be cancelled first. This simplifies the numbers you have to work with.