Problem 19 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ 3\left(n^{2}+1\right)-8\left(n^{2}-1\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-5n^2 + 11\).

Step 1: Distribute Terms Inside Parentheses

First, apply the distributive property to each term in the expression. Multiply the coefficients outside the parentheses with each term inside the parentheses.Expression: \[3(n^2 + 1) - 8(n^2 - 1)\]Distributing gives:\[3 imes n^2 + 3 imes 1 - 8 imes n^2 + 8 imes 1\]This simplifies to:\[3n^2 + 3 - 8n^2 + 8\]

Step 2: Combine Like Terms

Now, look for and combine like terms in the simplified expression. Notice the terms with \(n^2\) and combine them, and also combine the constant terms.Expression:\[3n^2 + 3 - 8n^2 + 8\]Combine the \(n^2\) terms:\[(3n^2 - 8n^2) = -5n^2\]Combine the constant terms:\[3 + 8 = 11\]

Step 3: Write the Final Simplified Expression

The expression is now completely simplified. Write the final expression by combining the results from the previous step.Final Expression:\[-5n^2 + 11\]

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that allows us to simplify expressions by removing parentheses. It states that multiplying a single term by two or more terms inside a set of parentheses is the same as doing each multiplication separately and then adding the results. In mathematical terms, if you have an expression in the form \(a(b + c)\), you distribute \(a\) to both \(b\) and \(c\) separately. Thus, it becomes \(ab + ac\).

• For our exercise, \(3(n^2 + 1)\) becomes \(3 \times n^2 + 3 \times 1\).
• Similarly, \(-8(n^2 - 1)\) is distributed to provide \(-8 \times n^2 + 8 \times 1\).

This step is important as it transforms the expression entirely by eliminating parentheses, making it easier for the next steps in simplification.

Simplifying Expressions

Simplifying expressions involves breaking down complex expressions into simpler forms. After using the distributive property, the expression has become longer, but not necessarily more complicated. We now focus on reducing it to its simplest terms.Using our example, upon distribution, we had:- \(3n^2 + 3 - 8n^2 + 8\).In this form, each term in the expression can easily be viewed, which makes the process of simplification more straightforward. Simplifying here means getting rid of any unnecessary complexity, making each step easy to digest.

Combining Like Terms

Combining like terms is a fundamental skill in algebra that helps simplify expressions by merging terms with the same variable raised to the same power. To put it simply, it means you only add or subtract numbers with the same kind of term attached to them.

• In the expression, \(3n^2\) and \(-8n^2\) are like terms because they both have \(n^2\). These can be combined to give \(-5n^2\).
• The constant terms \(3\) and \(8\) are also like terms and can be added together to give \(11\).

By combining these like terms, we ultimately arrive at our simplified final expression: \(-5n^2 + 11\). This simplifies the expression as much as possible, making it clearer and easier to work with in further mathematical operations.