Problem 81
Question
77–84 ? 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
Verified Answer
The expression factors to \( \left(x^2+1\right)^{-1/2} \left(x^2+3\right) \).
1Step 1: Identify Common Factor
Examine the expression \( \left(x^{2}+1\right)^{1 / 2}+2\left(x^{2}+1\right)^{-1 / 2} \). Notice that the common factor is \( \left(x^{2}+1\right)^{-1/2} \), since it is present in both terms as the lowest power of \( \left(x^2+1\right) \).
2Step 2: Factor Out the Common Factor
Factor out \( \left(x^{2}+1\right)^{-1/2} \) from each term: \[ \left(x^{2}+1\right)^{1 / 2} = \left(x^{2}+1\right)^{-1/2} \cdot \left(x^2+1\right) \] and the second term is already \( 2 \cdot \left(x^2+1\right)^{-1/2} \).
3Step 3: Simplify the Expression After Factoring
Once the common factor is factored out, the expression becomes: \[ \left(x^2+1\right)^{-1/2} \Big( (x^2+1) + 2 \Big) \] which simplifies to \[ \left(x^2+1\right)^{-1/2} \left(x^2+3\right) \].
Key Concepts
Common FactorSimplifying ExpressionsAlgebraic Expressions
Common Factor
Understanding how to work with a common factor is a crucial skill when factoring expressions. A common factor is a component present in all terms of an expression. When working with algebraic expressions, you aim to simplify or factor completely by identifying and extracting this common part from each term.
- Consider the expression \( \left(x^{2}+1\right)^{1 / 2}+2\left(x^{2}+1\right)^{-1 / 2} \). Here, \( \left(x^2+1\right) \) appears in both terms. Looking for the lowest power across all terms will help determine the common factor.
- In this case, \( \left(x^{2}+1\right)^{-1/2} \) is the common factor as it represents the smallest exponent of the recurring component.
Identifying the correct common factor is essential as it allows for the expression to be rewritten in a simpler, more manageable form. This not only makes the math easier to handle, but also reveals more about the structure of the problem being solved.
- Consider the expression \( \left(x^{2}+1\right)^{1 / 2}+2\left(x^{2}+1\right)^{-1 / 2} \). Here, \( \left(x^2+1\right) \) appears in both terms. Looking for the lowest power across all terms will help determine the common factor.
- In this case, \( \left(x^{2}+1\right)^{-1/2} \) is the common factor as it represents the smallest exponent of the recurring component.
Identifying the correct common factor is essential as it allows for the expression to be rewritten in a simpler, more manageable form. This not only makes the math easier to handle, but also reveals more about the structure of the problem being solved.
Simplifying Expressions
The goal of simplifying expressions is to make them easier to understand and work with. By factoring out the common elements, the expression becomes less complex.
- After identifying the common factor in \( \left(x^{2}+1\right)^{1 / 2}+2\left(x^{2}+1\right)^{-1 / 2} \), you can extract \( \left(x^{2}+1\right)^{-1/2} \) from each of its terms.
- This process involves rewriting: \[\left(x^{2}+1\right)^{1 / 2} = \left(x^{2}+1\right)^{-1/2} \cdot \left(x^2+1\right)\] and noting that the second part, \( 2 \cdot \left(x^2+1\right)^{-1/2} \), is already factored.
By extracting the common factor, the expression is transformed into a simpler form: \[ \left(x^2+1\right)^{-1/2} \Big( (x^2+1) + 2 \Big) \], which simplifies further to \[ \left(x^2+1\right)^{-1/2} \left(x^2+3\right) \].
Simplifying a factorized expression streamlines it, making it easier to understand, solve, or further manipulate, particularly in algebra where complexity can quickly escalate.
- After identifying the common factor in \( \left(x^{2}+1\right)^{1 / 2}+2\left(x^{2}+1\right)^{-1 / 2} \), you can extract \( \left(x^{2}+1\right)^{-1/2} \) from each of its terms.
- This process involves rewriting: \[\left(x^{2}+1\right)^{1 / 2} = \left(x^{2}+1\right)^{-1/2} \cdot \left(x^2+1\right)\] and noting that the second part, \( 2 \cdot \left(x^2+1\right)^{-1/2} \), is already factored.
By extracting the common factor, the expression is transformed into a simpler form: \[ \left(x^2+1\right)^{-1/2} \Big( (x^2+1) + 2 \Big) \], which simplifies further to \[ \left(x^2+1\right)^{-1/2} \left(x^2+3\right) \].
Simplifying a factorized expression streamlines it, making it easier to understand, solve, or further manipulate, particularly in algebra where complexity can quickly escalate.
Algebraic Expressions
An algebraic expression involves numbers, variables, and arithmetic operations combined together to reflect a mathematical relationship. Factoring and simplifying these expressions is a fundamental part of algebra that helps to reveal the expression's underlying structure.
- The expression \( \left(x^{2}+1\right)^{1 / 2}+2\left(x^{2}+1\right)^{-1 / 2} \) serves as an example of how expressions can be complex at first glance. However, by identifying shared components and factors, the problem becomes more accessible.
- Techniques such as factoring out the common term, simplifying second powers, or re-organizing elements can drastically alter the presentation and interpretability of the expression.
Mastering the art of handling algebraic expressions prepares you to tackle a wide variety of mathematical problems. This knowledge is applicable in fields like engineering, physics, and economics, where equations often need simplifying to make complex scenarios understandable.
- The expression \( \left(x^{2}+1\right)^{1 / 2}+2\left(x^{2}+1\right)^{-1 / 2} \) serves as an example of how expressions can be complex at first glance. However, by identifying shared components and factors, the problem becomes more accessible.
- Techniques such as factoring out the common term, simplifying second powers, or re-organizing elements can drastically alter the presentation and interpretability of the expression.
Mastering the art of handling algebraic expressions prepares you to tackle a wide variety of mathematical problems. This knowledge is applicable in fields like engineering, physics, and economics, where equations often need simplifying to make complex scenarios understandable.
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