12.6 Multiple Like-Term Groups

Multiple Like-Term Groups covers how an expression containing several different sets of like terms is fully simplified by reordering and regrouping the terms by variable part, combining each group independently, and reconstructing a final expression equivalent to the original.

Preparing the Expression for Grouping

Term Reordering for Group Formation

Using the commutative property, the terms of the expression are reordered so that terms sharing the same variable part sit next to one another, making the distinct like-term groups visually and structurally apparent before any combination takes place.

Term Regrouping by Variable Part

Using the associative property, the reordered terms are grouped with parentheses according to their shared variable part, forming distinct clusters that separate the entire expression into its constituent like-term groups.

Combining Each Group on Its Own

Independent Coefficient Combination

Once the terms are grouped, each like-term group is combined entirely on its own, adding or subtracting only the coefficients within that single group, with no interaction between the combination happening in one group and the combination happening in another.

Linear and Quadratic Term Separation

When an expression contains both linear terms and squared terms of the same variable, the two sets are kept in entirely separate groups, since they are not like terms to one another, and each group is combined independently to produce one linear term and one squared term in the final result.

Multivariable Group Separation

When an expression contains terms with different combinations of multiple variables, each distinct multivariable part forms its own separate group, and terms are combined only within a group whose variable part, including every variable and exponent, matches exactly.

Constant Group Separation

Every constant term in the expression, regardless of which variable groups also appear, is placed into its own separate constant group and combined independently, since constants share an identical empty variable part unrelated to any of the variable terms.

Handling Terms with No Match

Unmatched Term Preservation

A term whose variable part does not match any other term in the expression remains unmatched, and it is carried forward into the final simplified expression exactly as it was, untouched by any combination step, since there is no other term available to combine it with.

Assembling the Result

Final Expression Reconstruction

After every like-term group has been combined independently and every unmatched term has been carried forward, the resulting single terms and unmatched terms are reassembled into one final expression, typically written with the highest-power terms first, followed by lower powers, and any constant term last.

Equivalent Expression Result

The fully simplified expression produced by this process always represents the identical value as the original expression for every possible substitution of its variables, since every step used to arrive at it — reordering, regrouping, and combining — preserves value at each stage; only its written form has changed, not the quantity it represents.