Problem 56 - Algebra for College Students

Question

Perform the indicated operations. $$\left(6 n^{2}-4\right)-\left(5 n^{2}+9\right)-(6 n+4)$$

Step-by-Step Solution

Answer

The simplified expression is \( n^2 - 6n - 17 \).

Step 1: Distribute Negative Sign

The expression is given as \( (6n^2 - 4) - (5n^2 + 9) - (6n + 4) \). Start by distributing the negative signs across the terms followed by each parenthesis, thus converting the expression to \( 6n^2 - 4 - 5n^2 - 9 - 6n - 4 \).

Step 2: Combine Like Terms

Now we combine like terms in the expression:- Combine \( 6n^2 \) and \(-5n^2 \) which yields \( n^2 \).- The \(-6n\) term stands alone.- Combine the constant terms: \(-4 - 9 - 4 = -17 \).This simplifies the expression to \( n^2 - 6n - 17 \).

Key Concepts

Distributing Negative Signs

In algebraic expressions, parentheses combined with subtraction require you to distribute the negative sign across the terms inside. This means you change the sign of each term inside the parentheses. Let's break it down with the example expression: \[ (6n^2 - 4) - (5n^2 + 9) - (6n + 4) \]Initially, you must distribute the negative sign to each term in both sets of parentheses that follow a minus sign. - First, take the negative of each term in the second parenthesis: - \( - (5n^2 + 9) \Longrightarrow -5n^2 - 9 \)- Then, distribute the negative sign to the third parenthesis: - \( - (6n + 4) \Longrightarrow -6n - 4 \)This process leaves you with the expression without any parentheses:- \( 6n^2 - 4 - 5n^2 - 9 - 6n - 4 \)Distributing negative signs correctly sets the stage for effectively simplifying expressions further.

Combining Like Terms

Combining like terms involves merging terms in an expression that have the same variable and the same power, or combining constant terms. In the expression\[ 6n^2 - 4 - 5n^2 - 9 - 6n - 4 \], look for terms that can be combined. Here are some key pointers:

• Look at the terms with \(n^2\). We have \(6n^2\) and \(-5n^2\). Combine these to get \(n^2\).


• The \(-6n\) term has no other like terms, so it remains unchanged.


• Finally, add up all the constant numbers, \(-4\), \(-9\), and \(-4\), to find \(-17\).

After combining these, your expression becomes \(n^2 - 6n - 17\). Combining like terms simplifies expressions by reducing them to fewer terms, making them easier to understand and work with.

Simplifying Expressions

Simplifying an expression involves making it as compact as possible while still being equal in value. After distributing negative signs and combining like terms, simplifying allows for a more clear and concise expression. Starting from the combined expression:\[ n^2 - 6n - 17 \]This expression is already simplified because:

• Each term is as reduced as possible with no further factorization needed.


• There are no like terms left to combine.


• Operators are accurately applied with no extra parentheses for further simplification.

The expression \(n^2 - 6n - 17\) represents the final simplified version. Simplification makes mathematical expressions easier to interpret and communicate, helping students and mathematicians alike work more efficiently.