Identifying Expressions that require using the Distributive Property for Simplification
Recall the Distributive Property:
When a number or variable (or both) is multiplied by a series of terms in parentheses, it is
multiplied by each of the terms in parentheses.
• There will be a number or term (or a negative sign which can be thought of as a -1) multiplied by
a series of terms. Of course the series of terms is in parentheses.
• You can identify a series of terms because you will see at least one + or - sign in the ( ).
Use Distributive Property
Do NOT use the Distributive Property
7(5x + 3y − 2)
7 ⋅ 5x + 3y − 2
(The 7 is not multiplied by the series of terms,
just the one term.)
5 − (4x − 1)
(4x − 1) − 5
(The 5 is being subtracted. In order to multiply
by -5 a multiplication symbol, of some form,
must be there.)
3x(7 − 8x)
3x + (7 − 8x)
(The 3x is being added to, not multiplied by,
the series of terms.)
• The terms inside the ( ) will not be separated by a multiplication or division sign.
Use Distributive Property
Do NOT use the Distributive Property
7(6x + 2)
7(6x ⋅ 2)
−9(8x − 4)
−9(8x ÷ 4)
• The series of terms will be multiplied by another term, a number or variable or both.
Use Distributive Property
Use Distributive Property?
9(4x − y + 3)
yes
1
(9 − 12x)
(9 − 12x) which is the same as
3
3
9 + (4x − y + 3)
yes, note both forms
no, this is just adding terms--the parenthesis can
be ignored when simplifying
• Sometimes the multiplier is written before the series within ( ), sometimes after.
Use Distributive Property
Use Distributive Property?
9(4x − y + 3)
yes
yes, it’s the same as the example above it
(9)(4x − y + 3)
yes, it’s the same as the example above it
(4x − y + 3)9
yes, it is the same as the example above it
(Commutative Property)
• Recall: a negative sign before the ( ) can be thought of as multiplying by a -1.
Example
Simplified form (using the Distributive Property)
−(4x − 1)
−4x + 1
• After the ( ), a − sign simply indicates subtraction (unless it is in parentheses itself or it has a
multiplication sign in front of it).
Use Distributive Property
Do NOT use the Distributive Property
11− (−x + 5)
This simplifies to:
11+ x − 5
= x+6
(−x − 5) − 11
No Distributing.
The parenthesis can be ignored.
−x − 5 − 11
= −x − 16
• If the series is being multiplied by a term that has a - sign in front of it, include that sign with the
term when distributing.
Expression
Simplified form
(using the Distributive Property)
−6x(5 − 3x)
−30x + 18x 2
10 − 4(−1+ 5y)
10 + 4 − 20y
= −20y + 14
9x − 5(−2 − 3x)
9x + 10 + 15x
= 24x + 10