Problem 22 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \(a^{3}\left(a^{2}+3\right)-\left(a^{5}+3 a^{3}\right)\)

Step-by-Step Solution

Answer

The result of the expression is 0.

Step 1: Expand and Simplify Inside the Parentheses

First, we need to expand the expression. We have an expression inside parentheses: \( a^3(a^2 + 3) \). Distribute \( a^3 \) to both terms inside the parentheses: \[ a^3(a^2) + a^3(3) = a^5 + 3a^3 \] The expression now is: \[ a^5 + 3a^3 - (a^5 + 3a^3) \]

Step 2: Apply the Subtraction Across the Terms

Now subtract the second part of the expression \( (a^5 + 3a^3) \) from the first part \( a^5 + 3a^3 \). Distribute the negative sign across the terms in the parentheses: \[-a^5 - 3a^3\] So the expression becomes: \[ a^5 + 3a^3 - a^5 - 3a^3 \]

Step 3: Combine Like Terms

Combine the like terms in the expression. We notice that each \( a^5 \) and \( 3a^3 \) will cancel out with its negative counterpart: \[ (a^5 - a^5) + (3a^3 - 3a^3) = 0 + 0 = 0 \] Therefore, the expression simplifies to: \[ 0 \]

Key Concepts

Simplifying Expressions

Simplifying expressions involves rewriting a complex expression in a simpler or more easily understandable form. The goal is to make calculations easier or to recognize the value of the expression at a glance. To simplify the expression in the given exercise, we must carefully follow a series of steps. First, we evaluate any parentheses by carrying out any operations inside them, if possible. Then we work outward, applying arithmetic operations and simplifying wherever we can.

• Sometimes, simplifying an expression involves combining like terms or reducing expressions with factors common to all terms.
• On some occasions, the task of simplifying may reveal that an expression actually evaluates to zero, as in the case of the given problem.

Simplifying helps to make the expression more manageable and prepares it for further manipulations or evaluations.

Distributive Property

The Distributive Property of multiplication over addition is a key concept used when simplifying expressions. This property states that a term outside the parentheses can be distributed to each term inside the parentheses. In the original exercise, the Distributive Property was used to expand the expression:
\[ a^3(a^2 + 3) = a^3 \cdot a^2 + a^3 \cdot 3 \]
Breaking this down, you distribute the term outside, \(a^3\), to both terms inside the parentheses:

• First, \(a^3 \cdot a^2 = a^{5}\).
• Second, \(a^3 \cdot 3 = 3a^{3}\).

This property is crucial for expanding polynomials and allows you to transform an expression into a simpler form. By simplifying through distribution, the expression becomes more straightforward, laying the groundwork for further steps like subtraction or combining like terms.

Combining Like Terms

Combining like terms is an essential step in simplifying expressions and involves merging terms that have the same variable raised to the same power. This process reduces the number of terms in an expression, making it simpler and more compact.
In the provided exercise, after distributing and expanding the expression, the next step was to subtract and combine like terms:

• The expression \(a^5 + 3a^3 - a^5 - 3a^3\) contains like terms.
• When combing the like terms, \(a^5\) and \(-a^5\) cancel out, as do \(3a^3\) and \(-3a^3\).

The process of combining these like terms shows us that everything cancels out, resulting in zero. This reveals an important mathematical identity—that sometimes expressions look complicated but simplify to nothing. Understanding this process helps one better grasp algebraic manipulations and reveals the inherent symmetry or balance in algebraic structures.