Simplification with parentheses

When we simplify expressions with parentheses, it means that we remove the parentheses by multiplying in or using the sign rules, and then we collect like terms.

Parentheses are used to control the order of operations, so it is important to know the rules for how to work with them.

 

 

Example 1: Multiply into parentheses

Simplify the expression:

 

$$ \large 3(x+2) $$

 

We multiply 3 into the parentheses:

 

$$ \large 3 \cdot x + 3 \cdot 2 $$

$$ \large 3x + 6 $$

 

 

Example 2: Several terms in front of parentheses

Simplify the expression:

 

$$ \large 2a + 4(a-3) $$

 

First we multiply 4 into the parentheses:

 

$$ \large 2a + 4a - 12 $$

 

Now we collect like terms:

 

$$ \large 6a - 12 $$

 

 

Example 3: Negative sign in front of parentheses

Simplify the expression:

 

$$ \large 5x - (2x+7) $$

 

A minus in front of the parentheses means we change the sign of all terms inside the parentheses:

 

$$ \large 5x - 2x - 7 $$

 

Now we collect like terms:

 

$$ \large 3x - 7 $$

 

 

Example 4: Several parentheses

Simplify the expression:

 

$$ \large (x+4) + (2x-3) $$

 

Here we can remove the parentheses directly, because there is nothing in front of them:

 

$$ \large x + 4 + 2x - 3 $$

 

Now we collect like terms:

 

$$ \large 3x + 1 $$

 

 

Example 5: Multiply two parentheses together

Simplify the expression:

 

$$ \large (x+2)(x+3) $$

 

We multiply each term in the first parentheses with each term in the second parentheses:

 

$$ \large x \cdot x + x \cdot 3 + 2 \cdot x + 2 \cdot 3 $$

$$ \large x^2 + 3x + 2x + 6 $$

 

Collecting like terms, we get:

$$ \large x^2 + 5x + 6 $$

 

 

Example 6: Minus in front of a whole parentheses with several terms

Simplify the expression:

 

$$ \large (2x-5) - (3x-7) $$

 

We remove the parentheses. Remember that the minus in front of the second parentheses changes the signs:

 

$$ \large 2x - 5 - 3x + 7 $$

 

Now we collect like terms:

 

$$ \large -x + 2 $$

 

 

Summary

When you simplify expressions with parentheses, remember:

 

• Multiply all terms into the parentheses if there is something in front.
• A minus in front of parentheses changes the signs of all terms.
• If there is nothing in front of the parentheses, they can be removed directly.
• Collect like terms when the parentheses are removed.

 

Simplification with parentheses is an important step in working with algebraic expressions and equations, because it simplifies the expression so you can more easily move on to the next step.