Problem 90 - Algebra and Trigonometry

Question

Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$ 3 x^{-1 / 2}\left(x^{2}+1\right)^{5 / 4}-x^{3 / 2}\left(x^{2}+1\right)^{1 / 4} $$

Step-by-Step Solution

Answer

The factorized expression is \(x^{-1/2}(x^2+1)^{1/4}(2x^2+3)\).

Step 1: Identify Common Factors

Start by looking for the common factors in both terms of the expression: \(3x^{-1/2}(x^2+1)^{5/4}\) and \(x^{3/2}(x^2+1)^{1/4}\). These terms consist of powers of \(x\) and \((x^2+1)\). The common factor in the \(x\) terms is the lowest power of \(x\), which is \(x^{-1/2}\) and in the \((x^2+1)\) terms is \((x^2+1)^{1/4}\).

Step 2: Factor Out the Common Factor

Factor out \(x^{-1/2}(x^2+1)^{1/4}\) from the original expression. This results in: \[ x^{-1/2}(x^2+1)^{1/4} \left[ 3(x^2+1)^{1} - x^2 \right] \]. This is done by dividing each term by the common factor \(x^{-1/2}(x^2+1)^{1/4}\).

Step 3: Simplify the Remaining Expression

Now, simplify the expression inside the brackets: \[ 3(x^2+1) - x^2 = 3x^2 + 3 - x^2 = 2x^2 + 3 \]. This simplifies to \(2x^2 + 3\).

Step 4: Write the Final Factorized Expression

Combine the common factor and the simplified expression in the brackets to get the completely factorized expression: \[ x^{-1/2}(x^2+1)^{1/4} (2x^2 + 3) \]. This is the final factorized form of the original expression.

Key Concepts

Understanding Common Factors

When factoring algebraic expressions, the first step is to identify the common factors. Think of common factors as the shared building blocks in each term of an expression. In mathematics, these could be numbers, variables, or entire expressions. In the given problem, we started with the terms \(3x^{-1/2}(x^2+1)^{5/4}\) and \(x^{3/2}(x^2+1)^{1/4}\). The common factor is what they share in terms of multiplication components.

The common factor in the powers of \(x\) is the term with the smallest exponent, which is \(x^{-1/2}\). For the expression \((x^2+1)\), the smallest power is \((x^2+1)^{1/4}\). Factoring by these common factors helps simplify the entire problem, making it easier to manipulate and evaluate.

Exploring Powers of x

The concept of powers in algebra helps us understand how expressions grow and change. Powers of \(x\) refer to the number of times \(x\) is multiplied by itself. This is written in the form \(x^n\), where \(n\) is the exponent. For example, \(x^{-1/2}\) is a negative fractional power, meaning it's the reciprocal of the square root of \(x\). This can be less intuitive, but following the rules of exponents, it can be understood better.

In the original expression, the powers of \(x\) in each term need to be simplified by extracting the common factor. By lowering all terms to a consistent base power, factoring becomes much simpler. This manipulation makes it easier to see which elements can be grouped together.

Breaking Down Algebraic Expressions

Algebraic expressions are combinations of symbols, numbers, and operations. They can include variables (like \(x\)), constants, and coefficients. In our problem, the expression includes both powers of \(x\) and powers of \((x^2+1)\), which are nested within another expression.

When dealing with such expressions, the goal is often to simplify through factoring. This involves:

• Identifying each part of the expression: coefficients (such as 3 and 1), variables with exponents (such as \(x^{3/2}\)), and compound elements \((x^2+1)\).
• Recognizing which parts can be factored or divided out, based on common factors.
• Realizing that simplifying allows for easier manipulation and understanding of the expression’s behavior.

Breaking expressions into manageable parts makes the process of solving algebra easier and assists in recognizing the structure and pattern within the math.

Simplification for Clarity

Once the common factor is factored out of the original expression, the next task is simplification. Simplification is reducing an expression to its simplest form so it can be easily used in further calculations or problem-solving.

In this exercise, after factoring out the common term \(x^{-1/2}(x^2+1)^{1/4}\), the remaining expression \( [3(x^2+1) - x^2] \) needs simplification. This step involves basic arithmetic, distributing terms, and combining like terms. By simplifying the terms inside the brackets, we ended up with \(2x^2 + 3\). This concise version of the expression has no unnecessary components or redundancies. The process of simplification aids in reducing complex expressions to their core, which is vital for clarity and practical use in any algebraic manipulation.