14.7 Like-Term Reduction within Composite Expressions

Like-Term Reduction within Composite Expressions describes how combining like terms is applied as the stage of simplification that follows expansion, scanning the full set of terms produced across an entire expression, matching every one by variable part, and reducing each matched group into a single term.

Finding Matches after Expansion

Variable-Part Matching after Expansion

Once every grouped sub-expression has been expanded, the resulting flat list of terms is scanned for matching variable parts across the whole expression, including terms that originated from different groups or that were already present before any expansion took place.

Independent Like-Term Groups

Every distinct variable part found among the terms of the composite expression forms its own separate like-term group, and each group is reduced independently of every other group, with no interaction between the combination happening in one group and the combination happening in another.

Combining Coefficients within Each Group

Signed Coefficient Combination

Within each like-term group, the coefficients are added or subtracted according to their signs, exactly as with any signed-number combination, producing a single coefficient that represents the entire group.

Constant-Term Reduction

The constant terms scattered throughout the composite expression, regardless of which expanded group they originated from, are gathered into their own group and combined into a single constant term using ordinary addition and subtraction.

Cancellation to a Zero Coefficient

When the coefficients within a like-term group sum to zero, the entire group vanishes from the expression, since a variable part multiplied by a coefficient of zero contributes nothing to the expression's value.

What Is Left Unreduced

Nonmatching Term Retention after Like-Term Reduction

A term whose variable part does not match any other term's variable part anywhere in the composite expression remains on its own in the final result, carried forward exactly as it emerged from expansion, since there is no partner term available for it to combine with.

Handling Several Variables at Once

Multivariable Term Reduction

When the composite expression involves more than one variable, each distinct combination of variables and exponents forms its own separate like-term group, and reduction proceeds group by group exactly as with a single-variable expression, requiring the entire variable part, not just one variable, to match before terms are combined.

Assembling the Final Result

Final Term Reconstruction

Once every like-term group has been reduced to a single term, or eliminated entirely if its coefficient reduced to zero, and every unmatched term has been retained, all of these pieces are assembled into the final simplified expression, typically ordered from the highest power of the variable down to the constant term, completing the reduction of the composite expression.