Problem 95 - Algebra and Trigonometry

Question

Factor the expression completely. (This type of expression arises in calculus in 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 factored form is \((x^{2}+3)^{-4/3}(\frac{1}{3}x^{2} + 3)\).

Step 1: Identify Common Factors

Both terms of the expression \[(x^{2}+3)^{-1 / 3}-\frac{2}{3} x^{2}(x^{2}+3)^{-4 / 3}\]share a common factor. Observe that the common base \((x^{2}+3)\) is raised to different powers in each term. Find the smallest power that can be factored out from both terms, which is \((x^{2}+3)^{-4/3}\).

Step 2: Factor Out the Common Factor

Factor out \((x^{2}+3)^{-4/3}\) from the expression:\[(x^{2}+3)^{-1 / 3}=(x^{2}+3)^{-4/3} \cdot (x^{2}+3)^{1}\]Rewrite the second term by factoring \((x^{2}+3)^{-4/3}\):\[\frac{2}{3} x^{2}(x^{2}+3)^{-4 / 3} = \frac{2}{3} x^{2} \cdot (x^{2}+3)^{-4/3}\]Now factor out \((x^{2}+3)^{-4/3}\):\[(x^{2}+3)^{-4/3}[(x^{2}+3)^{1} - \frac{2}{3}x^{2}]\]

Step 3: Simplify the Expression Inside the Brackets

The expression inside the brackets is \[(x^{2}+3)^{1} - \frac{2}{3}x^{2}\]. Simplify it:\[(x^{2}+3) - \frac{2}{3}x^{2} = \frac{3}{3}x^{2} + 3 - \frac{2}{3}x^{2}\]Combine like terms by subtracting the coefficients of \(x^2\):\[\left(\frac{3}{3}x^{2} - \frac{2}{3}x^{2}\right) + 3 = \frac{1}{3}x^{2} + 3\]

Step 4: Write the Final Factored Form

Substitute the simplified term back into the factored expression:\[(x^{2}+3)^{-4/3}(\frac{1}{3}x^{2} + 3)\]This is the completely factored form of the original expression.

Key Concepts

Common Factors

When working with algebraic expressions, identifying common factors is a fundamental skill. A common factor is an expression that is present in all the terms of a given algebraic expression. In the exercise provided, you have two terms:

• \((x^{2}+3)^{-1 / 3}\)
• \(-\frac{2}{3} x^{2} (x^{2}+3)^{-4/3}\)

Both terms share a common factor of \((x^{2}+3)^{-4/3}\). Even though the same base \((x^2 + 3)\) appears in both terms with different exponents, we can still factor out the smallest power. Extracting common factors simplifies expressions, reducing complexity and aiding in further algebraic manipulations or calculus applications.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. They can range from simple expressions like \(x + 2\) to more complex forms such as the one in this exercise: \((x^{2}+3)^{-1 / 3}-\frac{2}{3} x^{2}(x^{2}+3)^{-4 / 3}\).It's crucial to rewrite parts of an expression to expose any commonalities that allow for simplification. This step often involves factoring out the greatest common factor or rearranging terms to recognize them. In our exercise, after identifying \((x^{2}+3)^{-4/3}\) as the common factor, we restructured the terms inside the brackets, which ultimately bespeaks the power of simplification in algebraic manipulation.

Product Rule in Calculus

The product rule in calculus is used to differentiate products of two functions. The rule states that if you have functions \(u(x)\) and \(v(x)\), then the derivative of their product \(u(x)v(x)\) is:\[u'(x)v(x) + u(x)v'(x)\]While the exercise focuses on factoring an algebraic expression, the expression itself is derived from scenarios in calculus where the product rule might apply. Understanding the origin of such expressions is essential, as it illuminates the context they arise in calculus. Factoring here simplifies the differentiation process later, as you'll have a more manageable expression to work with before applying further calculus operations.