Problem 90
Question
85–90 ? Factor the expression completely. (This type of expression arises in calculus when using the “product rule.”) $$ \frac{1}{2} x^{-1 / 2}(3 x+4)^{1 / 2}+\frac{3}{2} x^{1 / 2}(3 x+4)^{-1 / 2} $$
Step-by-Step Solution
Verified Answer
\( x^{-1/2} (3x + 4)^{-1/2} (3x + 2) \) is the factored form.
1Step 1: Identify Common Factors
First, look for common factors in the terms. The expression is \( \frac{1}{2} x^{-1 / 2}(3 x+4)^{1 / 2} + \frac{3}{2} x^{1 / 2}(3 x+4)^{-1 / 2} \). Notice that each term has a factor of \( x \) and \( (3x+4) \) with different exponents.
2Step 2: Find the Greatest Common Factor
Determine the greatest common factor from each term. For \( x^{-1/2} \) and \( x^{1/2} \), the common factor is \( x^{-1/2} \). For \( (3x+4)^{1/2} \) and \( (3x+4)^{-1/2} \), the common factor is \( (3x+4)^{-1/2} \). Hence, the overall greatest common factor is \( x^{-1/2} (3x+4)^{-1/2} \).
3Step 3: Factor Out the Common Factor
Factor \( x^{-1/2} (3x+4)^{-1/2} \) out of the expression:\[( x^{-1/2} (3x+4)^{-1/2} ) \left( \frac{1}{2} (3x+4) + \frac{3}{2} x \right)\]Simplify the factorized expression within the parenthesis.
4Step 4: Simplify the Parenthesized Expression
Inside the parenthesis, simplify the expression:\[\frac{1}{2} (3x + 4) + \frac{3}{2} x = \frac{1}{2} \cdot 3x + \frac{1}{2} \cdot 4 + \frac{3}{2} x \]Combine like terms:\[= \frac{3}{2} x + 2 + \frac{3}{2} x = 3x + 2\]
5Step 5: Write the Fully Factored Expression
Combine steps above to write the fully factored expression:\[x^{-1/2} (3x + 4)^{-1/2} (3x + 2)\]This represents the completely factored form of the given expression.
Key Concepts
Common FactorsGreatest Common FactorProduct RuleSimplification
Common Factors
When you factor expressions, identifying common factors is the first step. Common factors are elements that appear in each term of an expression. These factors can be numbers, variables, or algebraic expressions.
Spotting them helps to simplify and transform an expression into a product of factors. For example, look at the expression \[\frac{1}{2} x^{-1 / 2}(3 x+4)^{1 / 2} +\frac{3}{2} x^{1 / 2}(3 x+4)^{-1 / 2}\]. The common factor for terms involving \(x\) is the lowest power common to both terms, which is \(x^{-1/2}\).
Similarly, for \((3x+4)\), the lowest power is \((3x+4)^{-1/2}\). Identifying these common factors simplifies the factoring process greatly.
Spotting them helps to simplify and transform an expression into a product of factors. For example, look at the expression \[\frac{1}{2} x^{-1 / 2}(3 x+4)^{1 / 2} +\frac{3}{2} x^{1 / 2}(3 x+4)^{-1 / 2}\]. The common factor for terms involving \(x\) is the lowest power common to both terms, which is \(x^{-1/2}\).
Similarly, for \((3x+4)\), the lowest power is \((3x+4)^{-1/2}\). Identifying these common factors simplifies the factoring process greatly.
Greatest Common Factor
Finding the greatest common factor (GCF) is a key part of simplifying expressions. The GCF is the highest factor that all the terms in an expression share.
It often includes both numerical coefficients and variables.When finding the GCF of \[\frac{1}{2} x^{-1/2} (3x+4)^{1/2}\] and \[\frac{3}{2} x^{1/2} (3x+4)^{-1/2}\],we identify \(x^{-1/2}\) as the GCF for the \(x\) terms and \((3x+4)^{-1/2}\) for the \((3x+4)\) terms. Therefore, the overall GCF in this expression is \(x^{-1/2} (3x+4)^{-1/2}\). This factorization is essential to simplifying the terms for easier manipulation.
It often includes both numerical coefficients and variables.When finding the GCF of \[\frac{1}{2} x^{-1/2} (3x+4)^{1/2}\] and \[\frac{3}{2} x^{1/2} (3x+4)^{-1/2}\],we identify \(x^{-1/2}\) as the GCF for the \(x\) terms and \((3x+4)^{-1/2}\) for the \((3x+4)\) terms. Therefore, the overall GCF in this expression is \(x^{-1/2} (3x+4)^{-1/2}\). This factorization is essential to simplifying the terms for easier manipulation.
Product Rule
The product rule emerges prominently in calculus when differentiating products of two functions. To factor expressions arising from product rule applications, understanding multiplication properties is vital.Given the expression: \[\frac{1}{2} x^{-1/2} (3x+4)^{1/2}\], we express each part to consider common factors extensively. The product rule helps split and factorize these types of expressions efficiently.
This exercise involves terms reminiscent of derivatives where functions intertwine as products.
This exercise involves terms reminiscent of derivatives where functions intertwine as products.
- Recognize the structure as a product of terms.
- Seek common factors as discussed to rewrite the expression neatly.
Simplification
The simplification process aims to make expressions easier and more straightforward by combining like terms and reducing fractions.
Once common factors are identified and factored out, we look at what's inside the parenthesis.Consider \[\frac{1}{2} (3x+4) + \frac{3}{2}x\].
Distribute and combine like terms:- Combine \(\frac{3}{2}x\) and \(\frac{3}{2}x\) to get \(3x\).- Add constants.Simplifying gives us \(3x + 2\), which is the cleaner form within the brackets.
Applying simplification effectively showcases the coherent solution derived from initial complex expressions. Reducing the expression further supports clearer mathematical analysis and applications.
Once common factors are identified and factored out, we look at what's inside the parenthesis.Consider \[\frac{1}{2} (3x+4) + \frac{3}{2}x\].
Distribute and combine like terms:- Combine \(\frac{3}{2}x\) and \(\frac{3}{2}x\) to get \(3x\).- Add constants.Simplifying gives us \(3x + 2\), which is the cleaner form within the brackets.
Applying simplification effectively showcases the coherent solution derived from initial complex expressions. Reducing the expression further supports clearer mathematical analysis and applications.
Other exercises in this chapter
Problem 89
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