0580

Algebraic Fractions

Algebra and Sequences

Clearing the denominators

  • The syllabus lists these under equations rather than under algebraic fractions, but the method rests on everything above
  • Two routes reach the same place: combine the fractions into one and then cross-multiply, or clear the denominators one at a time
  • Clearing them is usually faster, provided each denominator multiplies every term on both sides, including any term that was never a fraction
  • Bracket a denominator as you multiply by it, since it is a whole expression rather than a single number
  • Removing two different algebraic denominators normally leaves a quadratic, so gather everything on one side and solve it as usual
  • A value that would make a denominator zero cannot be a solution, so discard it if it appears
Exam tip

Put the common denominator, with both tops over it, on a line of its own: mark schemes award a mark for the denominator alone and another for the combined top, so those marks are there before any expanding. Bracket the second top before subtracting it — that sign slip is the commonest way to lose the last mark. Cancel whole brackets only, never single terms.

Worked example

Two algebraic denominators

Solve 4 ÷ (x + 2) + 3 ÷ (x − 3) = 1.

Solution:

  • Multiplying throughout by (x + 2) clears the first fraction: 4 + 3(x + 2) ÷ (x − 3) = (x + 2)
  • Multiplying throughout by (x − 3) clears what is left: 4(x − 3) + 3(x + 2) = (x + 2)(x − 3)
  • Expand the left: 4x − 12 + 3x + 6, which collects to 7x − 6
  • Expand the right: x² − x − 6
  • Bring everything to one side: 0 = x² − 8x
  • There is no constant term, so take out x: 0 = x(x − 8)
  • The solutions are x = 0 or x = 8
  • Check x = 0: 4 ÷ 2 + 3 ÷ (−3) = 2 − 1 = 1
  • Check x = 8: 4 ÷ 10 + 3 ÷ 5 = 0.4 + 0.6 = 1