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How to Expand Parentheses with Variables


When you’re working with both variables and parentheses, you need to follow a particular set of rules. These rules allow you to remove the parentheses to clean up your expression. In school, you’ll spend a good amount of time working with expressions like this, so make sure to pay close attention.

Rule

Numbers and Signs Multiplied by Parentheses

1.
A + in front of a parenthesis changes nothing inside the parentheses.
2.
A in front of a parenthesis changes all the signs inside the parentheses, so a + becomes a and a becomes a +.
3.
A positive factor multiplied by a parenthesis leaves the signs unchanged, but the terms inside the parentheses are multiplied by the factor outside.
4.
A negative factor multiplied by a parenthesis changes all the signs inside the parentheses, and the terms inside the parentheses are multiplied by the factor outside.

Here are a couple of examples illustrating the four rules above:

Example 1

Rule 1

Expand the parentheses:

+(x + 1) = x + 1

Expand the parentheses:

+(x 1) = x 1

Expand the parentheses:

+(x 1) = x 1

Expand the parentheses:

+(x + 1) = x + 1

Example 2

Rule 2

Expand the parentheses:

(x + 1) = + x +1 = x 1

(x + 1) = + x +1 = x 1

Expand the parentheses:

(x 1) = + x 1 = x + 1

(x 1) = + x 1 = x + 1

Expand the parentheses:

(x 1) = x 1 = x + 1

(x 1) = x 1 = x + 1

Expand the parentheses:

(x + 1) = x +1 = x 1

(x + 1) = x +1 = x 1

Example 3

Rule 3

2a (a + 3) = + 2a (+a) l + 2a ( + 3) = 2a2 + 6a

2a (a + 3) = + 2a (+a) l + 2a ( + 3) = 2a2 + 6a

Expand the parentheses:

2a (a 3) = + 2a (+a) l + 2a ( 3) = 2a2 6a

2a (a 3) = + 2a (+a) l + 2a ( 3) = 2a2 6a

Expand the parentheses:

2a (a 3) = + 2a (a) l + 2a ( 3) = 2a2 6a

2a (a 3) = + 2a (a) l + 2a ( 3) = 2a2 6a

Expand the parentheses:

2a (a + 3) = + 2a (a) l + 2a ( + 3) = 2a2 + 6a

2a (a + 3) = + 2a (a) l + 2a ( + 3) = 2a2 + 6a

Example 4

Rule 4

Expand the parentheses:

a (b + a) = a (+b) l a ( + a) = ab a2

a (b + a) = a (+b) l a ( + a) = ab a2

Expand the parentheses:

a (b a) = a (+b) l a ( a) = ab + a2

a (b a) = a (+b) l a ( a) = ab + a2

Expand the parentheses:

a (b a) = a (b) l a ( a) = ab + a2

a (b a) = a (b) l a ( a) = ab + a2

Expand the parentheses:

a (b + a) = a (b) l a ( + a) = ab a2

a (b a) = a (b) l a ( a) = ab a2

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How Do You Multiply Parentheses?