Problem 55 - Algebra for College Students

Question

Perform the indicated operations. $$\left(n^{2}-7 n-9\right)-(-3 n+4)-\left(2 n^{2}-9\right)$$

Step-by-Step Solution

Answer

The simplified expression is \( -n^{2} - 4n - 4 \).

Step 1: Remove Parentheses

First, we remove the parentheses to simplify the expression. The expression given is \( (n^{2} - 7n - 9) - (-3n + 4) - (2n^{2} - 9) \). By removing the parentheses, we have:\[ n^{2} - 7n - 9 + 3n - 4 - 2n^{2} + 9 \]

Step 2: Combine Like Terms

Now, we combine the like terms from the expression:- Combine the \( n^{2} \) terms: \( n^{2} - 2n^{2} = -n^{2} \)- Combine the \( n \) terms: \( -7n + 3n = -4n \)- Combine the constant terms: \( -9 - 4 + 9 = -4 \)Thus, the expression becomes:\[ -n^{2} - 4n - 4 \]

Step 3: Simplify the Expression

The expression \( -n^{2} - 4n - 4 \) is already in its simplest form, as all like terms have been combined.Thus, the result of the original expression is:\[ -n^{2} - 4n - 4 \]

Key Concepts

Combining Like Terms

As you work through polynomial operations, it's vital to understand 'combining like terms.' This process involves grouping terms in the expression that have the same variables with identical exponents. Let's break it down.

In the given expression, we first identify each type of term:

• Quadratic terms, like those with the variable squared, such as \(n^2\).
• Linear terms, which are just \(n\) (to the first power).
• Constant terms, which are numbers without variables.

To combine them, add or subtract coefficients. In this expression, \(n^2 - 2n^2\) becomes \(-n^2\) because you subtract 2 from 1.

Similarly, for our \(n\) terms, we found \(-7n + 3n\) which simplifies to \(-4n\). For constants, \(-9 - 4 + 9\) simplifies to \(-4\). By combining like terms, you merge similar items for a concise expression.

Simplifying Expressions

After combining like terms, the next crucial step is 'simplifying expressions.' This means reducing the expression to its most manageable form. It's about trimming away complexity where possible without changing the expression's value.

The given expression simplifies to \(-n^2 - 4n - 4\) after combining like terms. Notice here, there aren't any more like terms to combine; each element in this expression is unique concerning its degree or lack thereof. This is how you know your expression is simplified:

• All like terms are combined.
• There are no remaining parentheses.
• You can't further reduce expression without altering its value.

This is the goal of simplification; to clearly present math work in the neatest form, making it easier to utilize in further operations or evaluations.

Algebraic Expressions

Algebraic expressions form the backbone of algebraic studies, representing numbers through variables instead of fixed numbers. Understanding them is pivotal for tackling more complex algebraic problems.

An algebraic expression is composed of terms connected by operators (like addition and subtraction). Each term might include a variable, coefficient, or constant. In our case, expressions like \(n^2 - 7n - 9\) are broken down into:

• Variables: Symbols that represent unknown values. Here, \(n\) is our variable.
• Coefficients: Numbers multiplying the variables, such as the 1 in \(1n^2\) (though often left out).
• Constants: Numbers on their own, free of variables, such as \(-9\).

Algebraic expressions allow you to perform operations where numbers change. They're flexible, enabling the solving of various problems by substituting values for variables or keeping them general for broader applications. By mastering algebraic expressions, you are building a foundation for more advanced mathematical exploration.