13.6 Expressions with Multiple Distributed Groups

Expressions with Multiple Distributed Groups covers how to expand an expression containing more than one separately grouped and factored sub-expression, distributing each group's own factor independently before combining all the resulting terms together into one fully expanded expression.

Expanding Each Group on Its Own

Independent Expansion of Separate Groups

When an expression contains two or more grouped sub-expressions, each with its own outer factor, every group is distributed independently of the others, using only that group's own factor and its own internal terms, with no interaction between the distribution happening in one group and the distribution happening in another.

How the Groups Relate to Each Other

Added Distributed Groups

When two distributed groups are joined by addition, each group is expanded on its own first, and the resulting sets of terms from both groups are then combined together using addition, exactly as their groups were originally joined.

Subtracted Distributed Groups

When two distributed groups are joined by subtraction, each group is still expanded independently first, but the second group's entire set of resulting terms then has its signs flipped before being combined with the first group's terms, since subtracting a group means subtracting everything that group expanded into.

Different Factors on Different Groups

Different Factors Applied to Different Groups

The outer factor multiplying one group in an expression is entirely independent of the outer factor multiplying any other group, and each factor, whatever its sign, size, or type, is applied only to the terms of its own group during expansion.

Signed Factor before Each Group

Whether each group's own outer factor is positive or negative, and whether the operation joining that group to the rest of the expression is addition or subtraction, are two separate considerations: the group's own factor determines the sign of its internally generated terms, while the operation joining it to the rest of the expression determines whether that entire set of terms is added or subtracted as a whole.

Carrying Out the Expansion in Order

Expansion Order across Multiple Groups

When an expression contains several distributed groups, each one can be expanded in any order relative to the others, since expanding one group does not affect the terms of a different, separately grouped sub-expression; only after every group has been expanded are all the resulting terms combined together.

Terms Outside Any Group

Ungrouped Term Preservation

Any term in the original expression that is not part of a distributed group, standing on its own outside any parentheses, is carried forward into the final expanded expression completely unchanged, exactly as it appeared originally.

Producing the Final Result

Complete Multi-Group Expansion

A complete expansion of an expression with multiple distributed groups results in a single sum of individual terms, with every grouping symbol removed, every group's own factor correctly applied to every one of its internal terms, and every ungrouped term preserved exactly as it was, ready to be simplified further by combining any resulting like terms.