52.5 Rational Expression Addition

Rational Expression Addition involves combining fractions with variables by finding common denominators and simplifying the result.

Rational Expression Addition is the procedure for combining two rational expressions into a single rational expression through addition, requiring both expressions to share an identical denominator before their numerators can be added together, and relying on the constructed least common denominator whenever the two original denominators differ. It splits naturally into two cases depending on whether the denominators already match: a direct combination when they do, and a conversion step through the LCD when they do not.

Because addition of fractions fundamentally requires a shared denominator, this procedure treats obtaining that shared denominator as a necessary first stage whenever the two expressions do not already meet that condition.

When the Denominators Already Match

Like-Denominator Addition Case

When both rational expressions already share the identical denominator, no conversion step is needed, and the addition proceeds directly by combining the two numerators over that shared denominator.

\frac{a}{c} + \frac{b}{c}

Numerator Sum Formation

The two numerators are added together, forming a single combined numerator that will sit over the shared denominator.

a + b

Common Denominator Retention

The shared denominator itself is carried forward unchanged into the result, since addition of fractions with a common denominator only combines the numerators, leaving the denominator as it was.

\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}

When the Denominators Differ

Addition LCD Selection

When the two denominators are not identical, the least common denominator is constructed from both denominators, following the rational LCD construction process, before either numerator can be combined.

First Addend Conversion

The first rational expression is multiplied, both numerator and denominator, by whatever factor converts its original denominator into the constructed LCD.

\frac{1}{2 x} \cdot \frac{x}{x} = \frac{x}{2 x^{2}}

Second Addend Conversion

The second rational expression is converted the same way, multiplied by whatever factor its own original denominator requires to reach the same LCD.

\frac{1}{x^{2}} \cdot \frac{2}{2} = \frac{2}{2 x^{2}}

Combining the Converted Terms

Converted Numerator Combination

With both expressions now sharing the identical LCD as their denominator, their numerators are added together directly, following the same process used in the like-denominator case.

\frac{x}{2 x^{2}} + \frac{2}{2 x^{2}} = \frac{x + 2}{2 x^{2}}

Finalizing the Result

Rational Sum Reduction

The combined numerator over the shared denominator is factored if possible and checked for any factor shared with the denominator, cancelling any such shared factor exactly as in general rational expression simplification.

Addition Domain Carryover

The domain restriction for the sum includes every excluded value from both of the original denominators before conversion, since the LCD is constructed directly from those denominators and inherits every value that would have made either of them zero.

x \neq 0