Method guideAlgebra IIIntermediate11 min read

Adding and subtracting rational expressions

Factor denominators, build the least common denominator, rewrite every numerator, then combine while preserving restrictions.

\frac ab+\frac cd=\frac{ad+bc}{bd}

Method guide

How to recognize the method, run it, and know when it is the wrong choice.

Updated July 13, 2026

MethodStart here

Denominators must name the same-sized algebraic pieces before numerators can combine.

The LCD is built from factors

Factor each denominator and include every distinct factor at its highest required power. Multiplying full denominators always works but may create avoidable clutter.

Record all zeros from the original denominators before rewriting anything.

x^2-4=(x-2)(x+2)

Rewrite with equivalent fractions

Multiply numerator and denominator by the missing factor. The value does not change because the fraction is multiplied by one.

Use parentheses around an entire numerator when subtraction is involved so the minus sign reaches every term.

\frac1{x-2}=\frac{x+2}{(x-2)(x+2)}

Combine, then simplify

Once denominators match, add or subtract only the numerators and keep the common denominator. Factor the new numerator if possible.

A factor created after combination may cancel, but original restrictions still remain.

\frac A D-\frac B D=\frac{A-B}{D}

Worked exampleBuild a shared denominator

Simplify 2/(x − 1) − 3/(x + 2).

D=(x-1)(x+2),\quad x\ne1,-2

Choose the LCD and restrictions.

\frac{2(x+2)}D-\frac{3(x-1)}D

Supply each missing factor.

\frac{2x+4-3x+3}{D}

Distribute the subtraction through the second numerator.

\frac{7-x}{(x-1)(x+2)}

Combine like terms.

Result

\boxed{\frac{7-x}{(x-1)(x+2)},\quad x\ne1,-2}

Watch for

Common mistakes

Adding denominators along with numerators.
Forgetting to multiply a numerator by the missing factor.
Failing to distribute a subtraction across a grouped numerator.

Keep

Three takeaways

Factor denominators first.
Use the least common denominator.
Combine numerators only after rewriting.