Simplifying rational expressions

A rational expression is one algebraic expression divided by another. To simplify it, factor the top and the bottom, then cancel a factor they share (one that is not zero), just as 12/30 becomes 2/5.

Example 16.

• top: x² – 7x + 12 = (x – 3)(x – 4);
• bottom: 5x² + 5x – 100 = 5(x² + x – 20) = 5(x – 4)(x + 5);
• cancel x – 4: the result is (x – 3)/(5(x + 5)).

Only cancel factors, never terms. In (x + 3)/(x + 5) you cannot cross out the x's: they are added, not multiplied.

Watch for factors that differ only in sign. (6s – t)² = (t – 6s)², because squaring removes the sign. So (36s² – 12st + t²)/((t + 8s)(t – 6s)) = (t – 6s)/(t + 8s).

Use every identity you know: difference of squares, sums and differences of cubes, perfect squares.

Sometimes nothing cancels. Then the factorised form is the simplest form.