12.5 Like-Term Combination

Like-Term Combination is the process of merging a group of identified like terms into a single term by adding or subtracting their coefficients while keeping their shared variable part unchanged, justified by the reverse application of the distributive property.

Combining Coefficients

Coefficient Addition

When two or more like terms all carry positive coefficients, or when their signs are being combined through addition, the coefficients are added together numerically to produce the coefficient of the single combined term.

Coefficient Subtraction

When one like term's coefficient is being subtracted from another's, the two coefficients are combined by subtraction rather than addition, following the sign of each term exactly as it appears in the expression.

What Stays Fixed during Combination

Variable-Part Preservation

Throughout the process of combining like terms, the shared variable part is carried through unchanged into the final combined term; only the coefficient changes as a result of the addition or subtraction, since the variable part was already confirmed identical across every term in the group before combination began.

Reverse Distributive Property Interpretation

Combining like terms can be understood as the reverse of distribution: instead of a factor being distributed across a sum to produce separate terms, the shared variable part is factored back out of the like terms, leaving a sum of coefficients multiplying that single shared factor.

Sign of the Resulting Coefficient

Positive Combined Coefficient

When the combined coefficients sum to a positive value, the resulting single term carries that positive coefficient attached to the shared variable part, behaving exactly like any other term with a positive coefficient in the rest of the expression.

Negative Combined Coefficient

When the combined coefficients sum to a negative value, the resulting single term carries that negative coefficient, and this negative sign is treated as part of the term going forward, exactly as with any other negative term.

Special Resulting Coefficients

Combined Coefficient of One

When the combined coefficients sum to exactly one, the resulting term is written with the variable part alone, since a coefficient of one is conventionally left implied rather than written explicitly.

Combined Coefficient of Negative One

When the combined coefficients sum to exactly negative one, the resulting term is written with only a negative sign in front of the variable part, since a coefficient of negative one is conventionally shown by the sign alone rather than the digit.

Combined Coefficient of Zero

When the combined coefficients sum to exactly zero, the entire term vanishes from the expression, since a variable part multiplied by a coefficient of zero contributes nothing to the expression's overall value.

Combining Constants

Constant-Term Combination

Constant terms, having no variable part, are combined the same way as any other like-term group: their numerical values are added or subtracted according to their signs, producing a single constant term that replaces the original group within the simplified expression.