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About this video
Adding and subtracting algebraic fractions the long way, with every step shown: factorise the denominators, build the LCD, write one big fraction, multiply in with brackets, then factorise the top and cancel.
What it covers
- Factorise every denominator first (difference of squares, trinomials)
- Building the LCD: a repeated factor once, a squared bracket keeps its power
- One big fraction with the LCD at the bottom — left factorised
- Brackets after a minus sign
- Factorise the numerator and cancel a common bracket
Extra worked examples
✍️ Extra example: 2/(x − 3) + 3/(x + 3) − 12/(x² − 9)
- 1x² − 9 = (x − 3)(x + 3), so the LCD is (x − 3)(x + 3).Factorise first; each bracket is taken once.
- 2(2(x + 3) + 3(x − 3) − 12)/((x − 3)(x + 3))Multiply each term by the LCD: its own denominator cancels and the rest multiplies its numerator.
- 3(2x + 6 + 3x − 9 − 12)/((x − 3)(x + 3)) = (5x − 15)/((x − 3)(x + 3))Expand, then collect like terms. The bottom stays factorised.
- 4(5(x − 3))/((x − 3)(x + 3)) = 5/(x + 3)Common factor 5 on top; the (x − 3) brackets cancel.
✍️ Extra example: (x + 1)/x − (x − 2)/2x
- 1LCD = 2xx and 2x share the x — take it once.
- 2(2(x + 1) − (x − 2))/2xThe minus sits in front of a BRACKET — this is where most marks are lost.
- 3(2x + 2 − x + 2)/2x = (x + 4)/2xThe minus changes both signs inside its bracket.
Exercises to try
Work each one out, then click to reveal the answer.
- 1Simplify 3/x + 5/2x.
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LCD 2x: (6 + 5)/2x = 11/2x - 2Simplify 4/(x − 1) − 2/(x + 1).
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(4(x + 1) − 2(x − 1))/((x − 1)(x + 1)) = (2(x + 3))/((x − 1)(x + 1)) - 3Simplify (x + 3)/4 − (x − 1)/6.
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LCD 12: (3(x + 3) − 2(x − 1))/12 = (x + 11)/12 - 4Simplify 6/(x² − 9) + 1/(x + 3).
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(6 + (x − 3))/((x − 3)(x + 3)) = (x + 3)/((x − 3)(x + 3)) = 1/(x − 3) - 5Simplify 1/((x + 1)²) + 2/(x + 1).
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LCD (x + 1)² — the squared bracket keeps its power: (1 + 2(x + 1))/((x + 1)²) = (2x + 3)/((x + 1)²)
