Problem 87 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The simplified form of the expression is \(-2x^2 -9\)

Step 1: Removing inner parenthesis

Start by removing the innermost parentheses first, then moving outwards. Distribution property is applied here. It becomes \(2(3 x^{2}-5)-(8 x^{2}-4 +3)\) which simplifies to \(2(3 x^{2}-5)-8 x^{2}-1\)

Step 2: Removing outer parentheses

Remove the outer parentheses by multiplying each term inside the parentheses with 2. This results in \(6x^2 -10 -8x^2 +1\)

Step 3: Combining like terms

Combine terms that have the same variable and exponent. This gives \(-2x^2 -9\)

Key Concepts

Simplification

Simplification in algebra involves reducing an expression to its simplest form, which makes it easier to understand and work with. This process typically involves a series of steps such as removing parentheses, combining like terms, and sometimes factoring.
In the exercise, simplification begins with identifying terms within parentheses and then appropriately applying distribution or combining strategies.
It's important to carefully perform each arithmetic operation, ensuring the rules of algebra are respected. Correct simplification allows for easier solving of equations or expressions, enabling a straightforward analysis of the problem at hand.

Distribution Property

The Distribution Property is a fundamental aspect of algebra used to simplify expressions. It refers to distributing a multiplier across terms within parentheses. This property can be remembered using the phrase, "multiply everything inside the parentheses by the outside number."
For example, if we have an expression like 2(3x - 5), using the distribution property means you multiply both 3x and 5 by 2, resulting in 6x - 10.
Applying this property is crucial because it allows you to eliminate parentheses, turning a complex expression into something more manageable. It is always a good idea to double-check each distribution step to prevent errors.

Combining Like Terms

Once you have distributed all multipliers and removed any parentheses, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power.
For instance, terms like 6x^2 and -8x^2 can be combined because they both involve x squared. You simply add or subtract their coefficients, in this case, resulting in -2x^2.
While combining like terms, it is essential to only add or subtract terms with identical variables and exponents. This process simplifies the expression further by reducing it to fewer terms, which makes it easier to read and solve.