Problem 129 - Introductory and Intermediate Algebra for College Students 4th

Question

Explain how to simplify an algebraic expression in which a negative sign precedes parentheses.

Step-by-Step Solution

Answer

In order to simplify an algebraic expression where a negative sign precedes parentheses, distribute the negative sign to each term inside the parentheses and subsequently remove the parentheses.

Step 1: Identify the negative sign

Find the parentheses and the negative sign in front of it. For example, in the expression -1(a+2), it is clear that the parentheses follow a negative sign.

Step 2: Distribute the negative sign

Next is to distribute the negative sign to each term inside the parentheses. Take heed that this will change the sign of each term in the parentheses. For instance, -1(a+2) would become -1*a + -1*2 which simplifies to -a - 2.

Step 3: Remove parentheses

In the final step, remove the parentheses. As the negative sign has been distributed to all elements within the parentheses, they are no longer necessary. So, our final simplified expression from the initial -1(a+2) is -a - 2.

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that allows us to simplify expressions by eliminating parentheses. It involves multiplying a term outside the parentheses by each term inside the parentheses. This ensures that each element within the parentheses is distributed by the factor outside, thereby maintaining equality.

A quick example might help to clarify. Consider the expression:

• 3(x + 4)

Using the distributive property, you'd multiply 3 by each term inside the parentheses: 3 × x and 3 × 4, which simplifies to 3x + 12.

This principle allows you to handle more complex algebraic functions smoothly and accurately, especially when dealing with negative signs and multiple terms inside the parentheses.

Negative Signs in Algebra

Negative signs can often be tricky for students to handle, especially when they appear in front of parentheses. When a negative sign precedes a parenthesis, it's essential to understand that it effectively "distributes" itself to each term inside that parenthesis. This changes the signs of all contained terms to their opposites.

For instance, the expression

• -1(a + 2)

means that the negative sign should affect both the 'a' and the '+2'. So, -1 multiplied by 'a' gives -a, and -1 multiplied by '+2' gives -2.

The simplified result would be - a - 2. A reliable way to simplify work involving negative signs is to remember to multiply the negative sign across each term inside the parentheses systematically. This will ensure accuracy and prevent common mistake areas related to sign errors.

Parentheses in Algebraic Expressions

In algebra, parentheses play a critical role in defining the order of operations within an expression. They indicate which operations should be performed first. However, when simplifying expressions, particularly with a negative sign outside the parenthesis, one of the goals is to remove them.

Once the distributive property is applied, parentheses may no longer be necessary. For example, in the expression

• -1(b + 3)

applying the negative sign through distribution transforms it to - b - 3.

The parentheses can then be removed because their purpose of defining the sequence of operations has been fulfilled. Parentheses ensure clarity when executing multiple operations, positioning them as vital tools for structuring and simplifying algebraic expressions.