Problem 20 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$4\left(n^{2}+3\right)+\left(n^{2}-7\right)$$

Step-by-Step Solution

Answer

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

Step 1: Distribute the Coefficient

Start by distributing the 4 across the terms inside the parentheses for the first expression. This means you will multiply 4 by each term inside:\[4(n^2 + 3) = 4 \cdot n^2 + 4 \cdot 3 = 4n^2 + 12\]

Step 2: Combine the Two Expressions

Write down the new expressions after distribution and include the second expression without any changes initially:\[4n^2 + 12 + (n^2 - 7)\]

Step 3: Remove Parentheses from the Second Expression

The second set of parentheses does not have a coefficient other than 1, so you can simply remove the parentheses:\[4n^2 + 12 + n^2 - 7\]

Step 4: Combine Like Terms

Combine the like terms, which are the terms with \(n^2\) and the constant numbers:- Combine \(4n^2\) and \(n^2\):\[4n^2 + n^2 = 5n^2\]- Combine the constants 12 and -7:\[12 - 7 = 5\]Resulting in:\[5n^2 + 5\]

Key Concepts

Like Terms

In algebra, "like terms" are terms that contain the same variable raised to the same power. Identifying and combining like terms is an essential skill when simplifying expressions. For instance, in the expression combining the terms:

• Both \(4n^2\) and \(n^2\) are like terms because they both contain the variable \(n\) raised to the power of 2.
• The constant terms \(12\) and \(-7\) are also considered like terms because they are numbers without variables.

By spotting these similarities, you can merge them together:

• For \(n^2\) terms: \(4n^2 + n^2 = 5n^2\)
• For constants: \(12 - 7 = 5\)

Once combined, our example simplifies to \(5n^2 + 5\). This simplification process is crucial because it makes the expression easier to understand and work with.

Distributive Property

The distributive property is a fundamental rule in algebra that helps simplify expressions. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.To distribute means to multiply the term outside the parentheses with each term inside the parentheses. For the expression \(4(n^2 + 3)\):

• Multiply 4 by \(n^2\), resulting in \(4n^2\).
• Multiply 4 by 3, giving us \(12\).

These multiplications yield the new expression: \(4n^2 + 12\). By utilizing the distributive property, you can eliminate parentheses and make it easier to combine like terms later on. Understanding and applying this property lets you simplify complex expressions more efficiently.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves using operations such as distribution and combining like terms to form a more straightforward expression.Here's a closer look at the process:

• First, use the distributive property to remove parentheses and expand expressions.
• Next, identify and combine like terms, such as variables with the same exponent and constant numbers.

In our example, after applying these steps, the expression became \(4n^2 + 12 + n^2 - 7\). Further simplification involved:

• Combining the variable terms: \(4n^2 + n^2\) to get \(5n^2\).
• Combining the constants: \(12 - 7\) to get 5.

The expression simplifies to \(5n^2 + 5\). This process is all about reducing the expression to its simplest form while maintaining its original value—a crucial ability for solving more complex problems and understanding deeper algebraic principles.