Algebraic fractions

An algebraic fraction has letters on the top or bottom. Simplify it like a number fraction: cancel common factors.

1. Factorise the top and the bottom fully.
2. Cancel any bracket or factor that is on both.

Example: simplify (x² − 9)/(x² + 5x + 6).

1. Top: x² − 9 = (x + 3)(x − 3)
2. Bottom: x² + 5x + 6 = (x + 3)(x + 2)
3. Cancel (x + 3): (x − 3)/(x + 2)

Use the same rules as number fractions, but factorise everything first so you can cancel.

1. To divide, flip the second fraction and multiply.
2. Factorise every top and bottom.
3. Cancel common factors, then multiply what's left.

Example: simplify (x + 2)/5 × 10/(x² − 4).

1. x² − 4 = (x + 2)(x − 2)
2. (x + 2)/5 × 10/((x + 2)(x − 2))
3. Cancel (x + 2), and 10 ÷ 5 = 2
4. 2/(x − 2)

Example: simplify 3x/(x + 1) ÷ 6x²/(x² − 1).

1. Flip and multiply: 3x/(x + 1) × (x² − 1)/(6x²)
2. x² − 1 = (x + 1)(x − 1)
3. Cancel (x + 1), and 3x/(6x²) = 1/(2x)
4. (x − 1)/(2x)

Algebraic fractions need a common denominator before you add or subtract, just like number fractions.

1. Use the product of the denominators as the common denominator.
2. Multiply each top by what its bottom was multiplied by.
3. Add or subtract the tops, then simplify.

Example: write 2/(x + 1) + 3/(x − 2) as a single fraction.

1. Common denominator: (x + 1)(x − 2)
2. Tops: 2(x − 2) and 3(x + 1)
3. 2(x − 2) + 3(x + 1) = 2x − 4 + 3x + 3 = 5x − 1
4. (5x − 1)/((x + 1)(x − 2))

When subtracting, put the second top in brackets so the minus sign applies to every term in it.

1. Multiply every term by all the denominators, so the fractions disappear.
2. Expand and rearrange (often to a quadratic = 0).
3. Solve, and check each answer in the original equation.

Example: solve 3/(x + 1) + 2/x = 2.

1. Multiply every term by x(x + 1): 3x + 2(x + 1) = 2x(x + 1)
2. 5x + 2 = 2x² + 2x
3. Rearrange: 2x² − 3x − 2 = 0
4. Factorise: (2x + 1)(x − 2) = 0
5. x = −½ or x = 2

Check x = 2: 3/3 + 2/2 = 1 + 1 = 2.