adding rational expressions

Adding rational expressions is the polynomial version of adding plain fractions, and it runs into the same requirement: you can only add fractions that share a denominator. Just as 1/6 + 1/4 forces you to rewrite both over 12, adding two rational expressions forces you to rewrite both over a common bottom.

The steps are: find a common denominator (the least common denominator is the tidiest choice), rewrite each expression with that denominator by multiplying top and bottom by the missing factors, add the numerators while keeping the common denominator, and finally simplify. Subtraction is identical, except you distribute the minus sign carefully across the entire second numerator.

When the denominators are already the same, you skip straight to combining numerators — that is the easy case. The work is almost entirely about engineering a shared denominator; once you have it, the addition itself is a single line.