52.2 Rational Expression Multiplication

Rational Expression Multiplication is the procedure for combining two rational expressions into a single rational expression by multiplying their numerators together and multiplying their denominators together, then simplifying the resulting product by cancelling any factors shared between the combined numerator and denominator. It mirrors the rule for multiplying ordinary numerical fractions, extending it directly to polynomials, and it benefits significantly from factoring each expression before combining them, since factors that would be difficult to spot after multiplying are often obvious beforehand.

Because cancellation can occur between the numerator of one factor and the denominator of either factor, factoring first rather than multiplying first typically produces a much simpler path to the fully reduced result.

Preparing Both Expressions

Numerator Pair Factoring

Both numerators, one from each rational expression being multiplied, are factored completely, using whatever combination of factoring techniques their structure requires, exactly as in rational expression factoring preparation.

Denominator Pair Factoring

Both denominators are factored completely the same way, producing a factored form for each of the four polynomials involved, two numerators and two denominators, before any multiplication is carried out.

Finding Cancellations Before Multiplying

Product-Stage Shared Factor Detection

With all four polynomials in factored form, every factor appearing in either numerator is compared against every factor appearing in either denominator, including cross-comparisons between the numerator of one expression and the denominator of the other.

Product-Stage Factor Reduction

Every matching pair of factors identified across the four polynomials is cancelled, one copy from a numerator against one copy from a denominator, before the remaining factors are multiplied together.

Combining What Remains

Reduced Numerator Product

Whatever numerator factors remain after cancellation from both original numerators are multiplied together, forming the numerator of the final product.

Reduced Denominator Product

Whatever denominator factors remain after cancellation from both original denominators are multiplied together the same way, forming the denominator of the final product.

Rational Product Assembly

The reduced numerator and reduced denominator are written together as the single simplified rational expression representing the completed multiplication.

Tracking the Domain

Product Domain Carryover

The domain restriction for the final product combines every excluded value from both original denominators, since a value making either original denominator zero would have made one of the two original rational expressions undefined before multiplication ever took place; this combined restriction is stated alongside the reduced product exactly as with any other rational expression simplification.

Even though the fully reduced product x − 2 shows no denominator at all, both excluded values from the original two denominators must still be carried forward and stated as part of the complete result.