Problem 28 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ -4\left(n^{2}+3\right)-\left(2 n^{2}-7\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-6n^2 - 5\).

Step 1: Expand the Expressions

Start by distributing the -4 across the terms inside the first parentheses. This turns \(-4(n^2 + 3)\) into \(-4n^2 - 12\). Next, distribute the negative sign across the second parenthesis, turning \(-(2n^2 - 7)\) into \(-2n^2 + 7\). So, the expression becomes \(-4n^2 - 12 - 2n^2 + 7\).

Step 2: Combine Like Terms

Identify and combine the like terms in the expression. The terms \(-4n^2\) and \(-2n^2\) are like terms because they both contain \(n^2\). Adding them gives \(-6n^2\). Then, combine the constant terms \(-12\) and \(+7\), giving you \(-5\).

Step 3: Write the Simplified Expression

Substitute the combined terms back into the expression. The simplified expression becomes \(-6n^2 - 5\).

Key Concepts

Distributive Property

The distributive property is a handy tool in algebra that allows us to remove parentheses from expressions. When you apply this property, you multiply the term outside the parenthesis by every term inside the parenthesis. For instance, in the expression \(-4(n^2 + 3)\), you distribute the \(-4\) to both \(n^2\) and \(+3\). This gives us \(-4n^2 - 12\).

• The coefficient \(-4\) is multiplied with \(n^2\) to get \(-4n^2\).
• The same coefficient \(-4\) is also multiplied with \(+3\) to yield \(-12\).

For the expression \(-(2n^2 - 7)\), distributing a negative sign is just as important. Here, the negative sign acts like \(-1\), which you multiply by each term inside the parentheses:

• \(-(2n^2) = -2n^2\)
• \(-(-7) = +7\)

By applying the distributive property, we transform complex expressions into simpler ones that are easier to handle.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. In our expression \(-4n^2 - 12 - 2n^2 + 7\), we need to identify and combine them:

• \(-4n^2\) and \(-2n^2\) are like terms because both have the variable \(n^2\). Adding these terms gives us \(-6n^2\).
• The constants, \(-12\) and \(+7\), are also like terms. Combining them yields \(-5\).

By replacing the expression with the combined terms, you achieve a simpler form: \(-6n^2 - 5\). This step ensures that expressions are succinct and comparable, making it easier to analyze or solve.

Negative Numbers in Algebra

Handling negative numbers is a fundamental algebraic skill. When dealing with expressions that have negative signs, careful attention is needed:

• Negative signs in front of parentheses, like in \(-(2n^2 - 7)\), mean the sign applies to each term inside. Thus, every term changes its sign.
• When you see subtraction like \(-4n^2\), think of it as adding a negative: so, \(-4n^2\) is \(+ (-4n^2)\).

This approach keeps the operations clear and avoids mistakes in further simplifications. In our problem, converting expressions through the distributive property with negative factors, like expanding to \(-4n^2 - 12\) and turning \(-7\) to \(+7\), demonstrates the meticulous use of negative numbers. By understanding these transformations, you’ll be more comfortable working with complex expressions in the future.