13.3 Negative Outer Factor Expansion

Negative Outer Factor Expansion applies the distributive property when the multiplying factor outside the group is negative, requiring the sign of every term inside the group to flip during distribution, including the special case where the outer factor is simply negative one.

Distributing a Negative Factor

Negative Factor Applied to a Sum

When a negative factor multiplies a group containing terms joined by addition, the factor is multiplied by each term separately as usual, but because the factor itself is negative, every resulting product carries a sign opposite to the sign the term originally had inside the group.

Negative Factor Applied to a Difference

When a negative factor multiplies a group containing a difference, each term is again multiplied by the negative factor individually, and the sign flip converts the original subtraction inside the group into an addition in the expanded result.

The Special Case of Negative One

Negative One as an Outer Factor

When the outer factor is exactly negative one, distribution simply flips the sign of every term inside the group with no change to any term's magnitude, since multiplying by negative one only reverses sign and leaves size unaffected.

Negated Group Expansion

A grouped expression preceded only by a negative sign, with no visible coefficient, is understood to have an implied outer factor of negative one, so expanding it means flipping the sign of every term inside exactly as with any other negative-one distribution.

Applying the Sign Flip Consistently

Sign Change across Every Grouped Term

Regardless of how many terms the group contains, every single one of them has its sign flipped when the outer factor is negative; no term inside the group is exempt from this sign reversal, since each one is multiplied by the same negative factor.

Negative Factor over Several Terms

When a negative factor multiplies a group with more than two terms, each term's sign is flipped individually according to its own original sign, producing an expanded expression where addition and subtraction have exchanged places throughout compared to the original group.

Handling Multiple Negative Signs

Consecutive Negative Signs during Distribution

When a term inside the group is already negative and it is multiplied by a negative outer factor, the two negative signs combine to produce a positive result for that particular term, following the same sign rule as multiplying any two negative numbers, even while every other positive term in the group becomes negative.

Confirming the Expansion

Negative-Factor Expansion Verification

A completed negative-factor expansion can be verified by substituting a chosen numerical value for any variables into both the original factored expression and the expanded expression, confirming the two results match; a mismatch typically indicates that the sign flip was applied to only some of the terms inside the group rather than every one of them.