Algebraic Fractions

Introduction:

Algebraic fractions are simply fractions in which the numerator (top part) and/or the denominator (bottom part) are algebraic expressions.

Key Concepts:

An algebraic fraction is equivalent to division. When divided by another algebraic fraction, you multiply by the reciprocal (flip the fraction around).
When adding or subtracting algebraic expressions, you need a common denominator, just like with regular fractions. If the denominators are the same, you can add or subtract the numerators; if they aren’t, you need to find the least common multiple of the two denominators.

Simplifying Algebraic Fractions:

An algebraic fraction can be simplified by factoring the numerators and denominators and cancelling out common factors.

Multiplying Algebraic Fractions:

When multiplying algebraic fractions, you simply multiply the numerators together and the denominators together.

Dividing Algebraic Fractions:

To divide by an algebraic fraction, you multiply by its reciprocal (swap the numerator and the denominator).

Adding and Subtracting Algebraic Fractions:

To add or subtract algebraic fractions, the fractions must have the same denominator. If they do not, you must find the least common denominator and adjust the fractions accordingly.

Using Algebraic Fractions in Equations:

Algebraic fractions can be used in equations, just like numbers. You can solve these equations using similar methods to those used to solve standard linear or quadratic equations.
The first step to solving an equation with algebraic fractions is to eliminate the fractions by multiplying through by a common multiple.

Roots of Algebraic Fractions:

To find the roots of an algebraic fraction, set the numerator equal to zero and solve. These are the values for which the algebraic fraction is undefined.

Misconceptions and Problems:

A common mistake is to cancel terms across adding or subtracting eg. (x+y)/y does not simplify to x. Only factors (terms separated by multiplying or dividing) can be cancelled.
Care must be taken with negative signs. Remember, dividing or multiplying by a negative flips the sign of the inequality.
Some fractions may seem to have a common denominator, but closer inspection reveals they do not. Always simplify each fraction independently before comparing denominators.