52.6 Rational Expression Subtraction

Rational Expression Subtraction is the procedure for combining two rational expressions into a single rational expression through subtraction, following the same shared-denominator requirement as addition, but requiring particular care in distributing the subtraction sign across every term of the second numerator, called the subtrahend, when that numerator contains more than one term. It parallels rational expression addition closely, differing mainly in this sign-distribution step, which is a common source of error if the subtrahend's numerator is treated as a single unbroken quantity instead of a polynomial whose every term must receive the negative sign.

Because subtracting a polynomial numerator means subtracting every one of its terms individually, this procedure treats the sign distribution as a distinct, deliberate step rather than an automatic consequence of writing a subtraction symbol.

When the Denominators Already Match

Like-Denominator Subtraction Case

When both rational expressions already share the identical denominator, the subtraction proceeds directly, without needing to construct a least common denominator first.

Subtracted Numerator Grouping

The numerator being subtracted, particularly when it consists of more than one term, is enclosed in parentheses as a single grouped quantity before the subtraction sign is distributed, making explicit that the entire expression, not just its first term, is being subtracted.

Complete Numerator Sign Distribution

The negative sign of the subtraction is distributed across every single term of the grouped numerator, flipping the sign of each term individually, exactly as with the distributive property applied to any subtracted polynomial.

When the Denominators Differ

Subtraction LCD Selection

When the two denominators are not identical, the least common denominator is constructed from both denominators, following the same rational LCD construction process used for addition.

Minuend Equivalent Conversion

The first rational expression, called the minuend, is converted to an equivalent expression over the constructed LCD, multiplying its numerator and denominator by whatever factor is required.

Subtrahend Equivalent Conversion

The second rational expression, called the subtrahend, is converted the same way over the same LCD, and its numerator is grouped in parentheses in preparation for the sign distribution that follows.

Combining the Converted Terms

Converted Numerator Difference

With both expressions now sharing the identical LCD, the subtrahend's converted numerator is subtracted from the minuend's converted numerator, with the sign distribution step applied fully across every term of that subtrahend numerator.

Finalizing the Result

Rational Difference Reduction

The combined numerator, after sign distribution and like-term reduction, 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.

Subtraction Domain Carryover

The domain restriction for the difference includes every excluded value from both original denominators before conversion, carried forward through the LCD exactly as in rational expression addition.