Problem 87 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Factored expression: \((x^2 + 1)^{-1/2}(x^2 + 3)\).

Step 1: Identify the Common Factor

First, we observe that both terms in the expression share a common factor, which is \((x^2 + 1)^{-1/2}\). Specifically, the lowest power between \((x^2 + 1)^{1/2}\) and \(2(x^2 + 1)^{-1/2}\) is \((x^2 + 1)^{-1/2}\). This will be factored out from the entire expression.

Step 2: Factor Out the Common Term

Factor \((x^2 + 1)^{-1/2}\) out of each term. For the first term, \((x^2 + 1)^{1/2} \div (x^2 + 1)^{-1/2} = (x^2 + 1)^{1}\). For the second term, \(2(x^2 + 1)^{-1/2} \div (x^2 + 1)^{-1/2} = 2\). Thus, when we factor out \((x^2 + 1)^{-1/2}\), the expression becomes \((x^2 + 1)^{-1/2}((x^2 + 1) + 2)\).

Step 3: Simplify the Expression Inside the Parentheses

Add the terms inside the parentheses: \((x^2 + 1) + 2 = x^2 + 3\). This simplifies our expression to \((x^2 + 1)^{-1/2}(x^2 + 3)\).

Step 4: Final Expression

The completely factored form of the original expression is \((x^2 + 1)^{-1/2}(x^2 + 3)\). This represents the simplest form by factoring out the lowest power of the common factor.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In the world of algebra, letters and symbols are used to represent numbers and quantities in formulas and equations. This allows for generalization and abstraction, making it easier to solve real-world problems. The main concepts of algebra include:

• Variables: Symbols that represent unknown values.
• Expressions: Combinations of variables, numbers, and operations.
• Equations: Statements that assert the equality of two expressions.

In solving algebraic problems, it's crucial to follow order of operations and understand properties of operations, such as distributive, associative, and commutative laws. In our exercise, algebra is applied when factoring expressions by recognizing and manipulating common factors and powers.

Common Factors

Factoring involves identifying common factors, which are elements shared by terms in an expression. This technique simplifies complex expressions by reducing them to more manageable forms.In our problem, the expression \((x^2 + 1)^{1/2} + 2(x^2 + 1)^{-1/2}\) was challenging due to different powers. The key was to find the lowest common power, which in this case, was \((x^2 + 1)^{-1/2}\).The steps involved included:

• Identifying the common base \((x^2 + 1)\) in both terms.
• Recognizing the lowest power, i.e., \(-1/2\).
• Factoring it out, simplifying the expression.

By factoring out the common factor, the expression was simplified, making it easier to manage through the remainder of the steps. Recognizing common factors is a valuable skill in algebra, as it helps in simplifying expressions and solving equations efficiently.

Simplification

Simplification is a fundamental aspect of algebra where expressions are rewritten in their simplest form. This involves performing operations and reducing expressions wherever possible. Simplification makes it easier to understand and work with mathematical problems because it streamlines them.In the given exercise, after factoring out the common factor \((x^2 + 1)^{-1/2}\), the expression inside the parentheses was \((x^2 + 1) + 2\). This expression was further simplified by adding the constants, resulting in \(x^2 + 3\).The process of simplification in this scenario included:

• Combining like terms.
• Ensuring expressions are in their minimal form for easier interpretation.
• Reducing potential errors in mathematical operations.

The final simplified expression, \((x^2 + 1)^{-1/2}(x^2 + 3)\), represents the most concise version of the original problem, demonstrating how powerful simplification can be in making algebraic expressions more approachable.