Manipulating Algebraic Fractions

# Manipulating Algebraic Fractions

GCSE(H),

An algebraic fraction is a fraction with an unknown. Simplifying an algebraic fraction involves determining common factors and dividing the numerator and the denominator by those factors.

Given frac(x^2 + 8x + 16)(x^2 - 16).

Factorise the numerator and the denominator to get frac((x + 4)(x + 4))((x + 4)(x - 4)).

The (x + 4) term is common to both the numerator and the denominator. Simplify to frac(x + 4)(x - 4).

When calculating with algebraic fractions, the same rules apply as they do for arithmetic fractions: when adding and subtracting, the denominators must be the same; multiplying involves multiplying the numerators together and multiplying the denominators together; and dividing involves inverting the second fraction, then multiplying the two fractions.

For example, simplify 1/(a + 4) + 3/(a - 4)

= frac((a - 4))((a + 4)(a - 4)) + frac(3(a + 4))((a - 4)(a + 4))

= frac(a - 4 + 3a + 12)((a + 4)(a - 4))

= frac(4(a + 2))((a + 4)(a - 4))

## Examples

1. Simplify frac((a^2 + 4a + 4))((a^2 - 4)).

Answer: frac((a + 2))((a - 2))

frac((a^2 + 4a + 4))((a^2 - 4)) = frac((a + 2)(a + 2))((a + 2)(a - 2)) = frac((a + 2))((a - 2))

2. Calculate frac((x - 3))((x + 3)) x frac((4x + 12))((2x - 6)).

frac((x - 3))((x + 3)) x frac((4x + 12))((2x - 6)) = frac(4(x - 3)(x + 3))(2(x + 3)(x - 3)) = frac(4)(2) = 2