Multiplying, Dividing, Adding & Subtracting

The rules are identical to numeric fractions.

Multiplying: factor everything, cancel across the whole product, then multiply what is left.

Dividing: flip the second fraction and multiply.

Adding and subtracting: you need a common denominator. Factor each denominator, then build the LCD by taking every distinct factor at its highest power.

For 1/(x² − 4) + 3/(x + 2): factor the first denominator to (x+2)(x−2), so the LCD is (x+2)(x−2). Multiply the second fraction by (x−2)/(x−2) and combine.

When subtracting, distribute the minus across the entire second numerator. That is the single most common error.

Multiplication and division follow (a/b)(c/d) = ac/bd and (a/b) ÷ (c/d) = ad/bc, with all cancellation done after complete factorisation. Division carries the extra restriction that the divisor's numerator must also be nonzero, since it becomes a denominator.

Addition requires the LCD, constructed from the factored denominators by taking each distinct irreducible factor to its highest occurring power — exactly the LCM procedure from prime factorisation, with irreducible polynomials in place of primes.

Subtraction requires the numerator of the second fraction to be treated as a grouped quantity: a/d − (b + c)/d = (a − b − c)/d. Failing to distribute the minus is the dominant error in the topic.

Restrictions accumulate across every step: any value making any denominator zero at any stage is excluded, including denominators that were eliminated by cancellation.

Complex fractions are handled either by simplifying numerator and denominator separately and then dividing, or by multiplying every term by the overall LCD, which is generally faster.