Problem 88 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression by removing parentheses and brackets. $$4\left(6 x^{2}-3\right)-\left[2\left(5 x^{2}-1\right)+1\right]$$

Step-by-Step Solution

Answer

The simplified expression is \(14x^{2} - 11\).

Step 1: Distribute 4

Distribute the 4 in the first term \(4(6 x^{2}-3)\) across the terms inside the parentheses: \(24x^{2} - 12\)

Step 2: Distribute 2

For the expression inside the brackets, first distribute 2 across the terms inside the parentheses: \(2(5x^{2} - 1)\) becomes \(10x^{2} - 2\). Adding 1 to this gives \(10x^{2} - 2 + 1 = 10x^{2} - 1\)

Step 3: Subtract the bracketed term

Combine the terms you've now obtained to simplify the expression: \(24x^{2} - 12 - [10x^{2} - 1]\). Remember to distribute the negative sign to each term in the brackets: \(24x^{2} - 12 - 10x^{2} + 1\)

Step 4: Combine like terms

Combine the similar terms: \(24x^{2} - 10x^{2} = 14x^{2}\) and \(-12 + 1 = -11\). Giving \(14x^{2} - 11\)

Key Concepts

Distributive Property

One of the foundational tools in simplifying algebraic expressions is the distributive property. This property is applied when you multiply a single term by two or more terms inside a set of parentheses. Consider the expression given: \( 4(6x^{2} - 3) \). The distributive property allows you to open up the parentheses by multiplying each term inside by 4.
Here's how it works:

• Multiply 4 by \( 6x^{2} \), resulting in \( 24x^{2} \).
• Multiply 4 by \(-3\), resulting in \(-12\).

Thus, the expression \( 4(6x^{2} - 3) \) simplifies to \( 24x^{2} - 12 \). This operation is essential because it helps in eliminating parentheses, making the expression easier to handle in subsequent steps.
Next, the distributive property is also applied to \( 2(5x^{2} -1) \) to get \( 10x^{2} - 2 \). Once you distribute thoroughly, any further calculations must follow.

Combining Like Terms

After using the distributive property, you'll often be left with similar terms, known as like terms, that can be combined for simplicity. Like terms have the same variable raised to the same power. In the given exercise, the terms \( 24x^{2} \) and \(-10x^{2} \) are like terms because they both contain \( x^{2} \).
To combine these like terms, you need to add or subtract their coefficients:

• Add the coefficients of \( x^{2} \): 24 and -10, which results in 14. Thus, the term \( 24x^{2} - 10x^{2} \) becomes \( 14x^{2} \).
• For the constant terms, combine \(-12\) and \(+1\) to get \(-11\).

By combining like terms, you're essentially simplifying the expression further, reducing it to fewer terms. This step is crucial as it often leads to a cleaner expression that offers better insight into the problem's structure.

Expression Simplification

Expression simplification involves both the application of the distributive property and the combination of like terms. It is the process of reducing an algebraic expression to its simplest form. In this specific exercise, you've started with a complex expression with multiple layers of operations: \( 4(6x^{2}-3) - [2(5x^{2}-1) + 1] \).
The goal is to systematically eliminate parentheses, combine what you can, and reorder terms if necessary to achieve clarity:

• After distributing within each set of parentheses and brackets, you had \( 24x^{2} - 12 \) and \( 10x^{2} - 1 \).
• You've then carefully handled the subtraction of bracketed terms, making sure to distribute the subtraction across all terms within the brackets: \( -[10x^{2} - 1] \).
• Finally, by combining like terms, namely, reductions in \( x^{2} \) terms and constants, the expression reached its simplest form: \( 14x^{2} - 11 \).

This process not only organizes the expression but also ensures it logically represents what's occurring mathematically, creating ease in interpretation and further use or manipulation.