Problem 89 - Algebra and Trigonometry

Question

85–90 ? Factor the expression completely. (This type of expression arises in calculus when using the “product rule.”) $$ \left(x^{2}+3\right)^{-1 / 3}-\frac{2}{3} x^{2}\left(x^{2}+3\right)^{-4 / 3} $$

Step-by-Step Solution

Answer

The expression is factored as \((x^2 + 3)^{-4/3} \left( \frac{1}{3}x^2 + 3 \right)\).

Step 1: Identify Common Factors

First, observe that both terms in the expression contain the factor \((x^2 + 3)^{-1/3}\). Hence, our goal is to factor this common term out of the expression.

Step 2: Factor Out the Common Term

Factor out \((x^2 + 3)^{-4/3}\) from both terms, giving:\[(x^2 + 3)^{-4/3} \left( (x^2 + 3)^{1} - \frac{2}{3}x^2 \right)\]

Step 3: Simplify Inside the Parentheses

Inside the parentheses, simplify by performing the subtraction:\[(x^2 + 3) - \frac{2}{3}x^2 = 1 \cdot x^2 - \frac{2}{3}x^2 + 3\]Combine like terms:\[\frac{3}{3}x^2 - \frac{2}{3}x^2 + 3 = \frac{1}{3}x^2 + 3\]

Step 4: Reconstruct the Factored Expression

With the simplification inside the parentheses, the fully factored expression is:\[(x^2 + 3)^{-4/3} \left( \frac{1}{3}x^2 + 3 \right)\]

Key Concepts

Calculus

In calculus, we often deal with complex expressions that involve derivatives and integrals. One common technique in calculus is the manipulation of algebraic expressions to simplify derivative calculations.
When performing calculus operations, understanding how to factor expressions can make solving problems much easier. It's like breaking down a big problem into smaller, more manageable parts. By factoring expressions, we can see the common components shared between terms, which can lead to significant simplifications, especially when dealing with the product rule.

Product Rule

The product rule in calculus is essential for finding the derivative of products of two or more functions. This rule states that if we have two functions, say \( u(x) \) and \( v(x) \), the derivative of their product is calculated as:
\( (uv)' = u'v + uv' \).
Consider our initial expression: it closely relates to the kinds of expressions you'd encounter when applying the product rule. Recognizing parts of the expression that can be factored out simplifies both the derivative operation and further algebraic manipulations.

• Your aim is to recognize expressions that recur throughout a problem.
• Look for opportunities to simplify before or after applying the product rule.

Simplifying Expressions

Simplifying expressions is a critical skill in mathematics and calculus. It involves combining like terms, factoring, and reducing expressions to simpler forms. When faced with an expression like our original problem, identifying and extracting common factors is a vital first step.
This approach not only makes the expression manageable but also ensures that subsequent operations, such as differentiation or integration, are simplified. By simplifying expressions, we reduce the chance of errors and enhance the efficiency of problem-solving.

• Identify common factors across terms.
• Perform operations such as addition or subtraction step by step to combine like terms.
• Always aim for the simplest equivalent expression for ease of further manipulation.

Algebraic Expressions

Algebraic expressions form the backbone of most calculus problems. They consist of variables, numbers, and operations combined into terms that can be manipulated according to algebraic rules. Factoring these expressions is a crucial skill. You need to look for the greatest common factor (GCF) among terms to simplify them. Recognizing patterns such as squares, cubes, or binomials plays a key role.

• Learn the rules for identifying and factoring common components.
• Practice identifying and rewriting expressions to see potential simplifications.
• Mastering algebra helps in tackling advanced calculus problems as it lays the foundation for a deeper understanding of all mathematical operations.