Problem 64 - Algebra for College Students

Question

Simplify by removing the inner parentheses first and working outward. $$-\left(3 n^{2}-2 n+4\right)-\left[2 n^{2}-\left(n^{2}+n+3\right)\right]$$

Step-by-Step Solution

Answer

The expression simplifies to \(-4n^2 + 3n - 1\).

Step 1: Simplify the Inner Parentheses

Start by simplifying the content inside the innermost parentheses first: \(n^2 + n + 3\) stays unchanged as it's inside a subtraction. Now, subtract this expression from \(2n^2\), which is outside the brackets, giving:\[2n^2 - (n^2 + n + 3) = 2n^2 - n^2 - n - 3 = n^2 - n - 3\]

Step 2: Simplify the Brackets

Replace the expression inside the square brackets with the result from the previous step:\[-(3n^2 - 2n + 4) - [n^2 - n - 3]\] The expression inside the square brackets has already been simplified.

Step 3: Distribute the Negative Sign to the Brackets

Now, distribute the negative sign to each term inside the square brackets:\[-(n^2 - n - 3)\] becomes \[-n^2 + n + 3\]

Step 4: Simplify the Entire Expression

Place the newly formed expression in the original equation:\[-(3n^2 - 2n + 4) - n^2 + n + 3\]Now distribute the negative sign from the initial part:\[-3n^2 + 2n - 4 - n^2 + n + 3\]Combine like terms to simplify:\[-3n^2 - n^2 + 2n + n - 4 + 3 = -4n^2 + 3n - 1\]

Key Concepts

Polynomial Expressions

Polynomial expressions are combinations of variables and constants using addition, subtraction, and multiplication. A typical polynomial looks like this: \(3x^2 + 2x - 5\). The expressions involve terms, which are segments that are separated by plus or minus signs. These terms can include constants, variables, or variables raised to some power.
To work with polynomials, one must understand each part:

• Coefficients: These are the numbers in front of the variables, like the \(3\) in \(3x^2\).
• Variables: Usually represented by letters such as \(n\) or \(x\), they stand for unknown values.
• Exponents: These are the powers to which the variables are raised, such as the \(^2\) in \(x^2\).

Polynomial expressions can be simplified by combining like terms and reducing them into their simplest form. This is a crucial step in solving algebraic equations and inequalities.

Parentheses in Expressions

Parentheses control the order in which operations are performed in mathematical expressions. This is part of the broader concept known as the "Order of Operations," typically remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In expressions, parentheses indicate that the enclosed terms should be treated as a single unit and simplified first. For instance, in the expression \(- (3n^2 - 2n + 4)\), you first handle what's inside the parentheses before dealing with the negative sign outside.
In our exercise, by working from the innermost to the outermost parentheses or brackets, we ensure a systematic simplification of the expression. Always work on reducing within the parentheses to simpler terms before proceeding to the rest of the expression.

Like Terms

Like terms are terms in an expression that have identical variable parts, meaning they can be combined through addition or subtraction. For example, in the expression \(2n^2 + 3n^2\), both terms are like terms because they have the same variable part, \(n^2\).
When simplifying expressions, combining like terms is a crucial step:

• Make sure the terms have the exact same variable and exponent.
• Add or subtract the coefficients of these terms.

In our example, the expression \(-3n^2 - n^2\) simplifies to \(-4n^2\) by combining the coefficients \(-3\) and \(-1\). This process significantly reduces the complexity of polynomial expressions, making them easier to solve.

Negative Sign Distribution

Distributing a negative sign means applying it to each term inside a set of parentheses. This means each term will switch its sign from positive to negative or vice versa.
For instance, if we have \(- (n^2 - n - 3)\), distributing the negative sign results in \(-n^2 + n + 3\). Here are the steps you generally follow:

• Change the sign of every term in the parentheses.
• Maintain the expression's overall integrity by rewriting it with these new signs.

It's important because not correctly distributing the negative can lead to incorrect simplifications and solutions. This step can often be overlooked, but it is vital for accurately solving algebraic expressions and ensuring the simplification process is completed correctly.