Working with Algebraic Fractions
This topic covers the core skills of manipulating algebraic expressions. It involves 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.