Problem 82 - Algebra and Trigonometry

Question

77–84 ? Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$ x^{-1 / 2}(x+1)^{1 / 2}+x^{1 / 2}(x+1)^{-1 / 2} $$

Step-by-Step Solution

Answer

The expression factors to \(x^{-1/2}(x+1)^{-1/2}(2x + 1)\).

Step 1: Identify Common Factors

Examine both terms in the given expression: \(x^{-1/2}(x+1)^{1/2}\) and \(x^{1/2}(x+1)^{-1/2}\). The common factors are \(x\) and \(x+1\). Identify the lowest powers of these factors.The lowest power of \(x\) is \(x^{-1/2}\) and of \(x+1\) is \((x+1)^{-1/2}\).

Step 2: Factor Out the Lowest Powers

Factor the expression by taking out the lowest powers identified in Step 1:\[(x^{-1/2}(x+1)^{-1/2}) \left[ (x+1) + x \right] \]

Step 3: Simplify the Remaining Expression

Simplify the expression inside the bracket:\((x+1) + x = 2x + 1\).So the expression becomes:\[ (x^{-1/2}(x+1)^{-1/2}) (2x + 1)\]

Step 4: Rewrite the Factored Expression

The expression: \[(x^{-1/2}(x+1)^{-1/2})(2x+1)\] is now factored completely.Therefore, the final factored form is:\[ x^{-1/2}(x+1)^{-1/2}(2x + 1) \]

Key Concepts

Understanding Common Factors

When we talk about common factors in algebraic expressions, we're referring to terms or parts of an expression that appear in multiple components of an equation. These are the pieces that can be "factored out," or taken as a common element, to simplify the expression further. Recognizing common factors is the first step in the factoring process.
For instance, if you have an expression like \(x^{-1/2}(x+1)^{1/2} + x^{1/2}(x+1)^{-1/2}\), the common element is what you search for in both terms. Here, both terms contain the factors \(x\) and \(x+1\). The key is to find the lowest exponent of these factors across the terms, which are \(x^{-1/2}\) and \((x+1)^{-1/2}\) respectively.

• Identify terms common to all parts of the expression.
• Find the lowest power of those terms.

Recognizing and working through common factors can greatly simplify the process of manipulating algebraic expressions.

Simplifying Expressions

Simplifying expressions is a major skill in algebra that makes complex problems easier to manage. Once common factors are identified, the next step is to simplify the expression by factoring out these components.
After factoring out the common part in our example, the expression inside the parenthesis becomes much simpler:\[(x^{-1/2}(x+1)^{-1/2})\left[(x+1) + x\right]\]
Inside the brackets, \((x+1) + x\) simplifies to \(2x + 1\). This results in the expression: \[(x^{-1/2}(x+1)^{-1/2})(2x + 1)\]. This substitution makes it much more straightforward to understand and work with the expression.

• Combine like terms where possible.
• Simplify arithmetic operations inside the expression.

Simplifying as you factor can make challenging algebraic tasks much more approachable and less error-prone.

Exploring Algebraic Expressions

Algebraic expressions, like the one in this example, are composed of variables, coefficients, and mathematical operations. These expressions can represent almost anything and vary from basic to complex structures.
In the expression \(x^{-1/2}(x+1)^{1/2} + x^{1/2}(x+1)^{-1/2}\), each term is a product of powers of different variables. Working with algebraic expressions involves both identifying like terms and operations, and then using strategies like factoring. This helps reveal underlying simplicity or solve equations.

• Understand the structure of every term in the expression.
• Apply operations like adding, multiplying, and factoring.

By focusing on the core properties of these expressions, students can unlock easier solving techniques and gain confidence in tackling more intricate mathematical problems.