/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 92 Factor: \((2 x+1)^{-\frac{1}{2}}... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 92

Factor: \((2 x+1)^{-\frac{1}{2}}\left(x^{2}+3\right)^{-\frac{1}{2}}-\left(x^{2}+3\right)^{-\frac{3}{2}} x(2 x+1)^{\frac{1}{2}}\)

Short Answer

Expert verified
\( (2x+1)^{-\frac{1}{2}}(x^2+3)^{-\frac{3}{2}}(-x^2 - x + 3) \)

Step by step solution

01

Identify common factors

Look for common factors in the given expression. Both terms have factors involving \((2x+1)\) and \(x^2+3\).
02

Extract common factors

Extract \((2x+1)^{-\frac{1}{2}}\) and \((x^2+3)^{-\frac{3}{2}}\) as common factors: \[ (2x+1)^{-\frac{1}{2}}(x^2+3)^{-\frac{3}{2}} \left[ (x^2+3) - x(2x+1) \right]. \]
03

Simplify the expression within brackets

Simplify within the brackets: \[ (x^2+3) - x(2x+1) = x^2 + 3 - 2x^2 - x = -x^2 - x + 3. \]
04

Write the fully factored expression

Combine the simplified inner expression with the common factors: \[ (2x+1)^{-\frac{1}{2}}(x^2+3)^{-\frac{3}{2}}(-x^2 - x + 3). \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Common Factors
To factor an algebraic expression, finding common factors is crucial. Common factors are elements that appear in each term of the expression. Identifying them allows us to group and simplify expressions more easily.

For example, in the expression above, both terms include \( (2x+1) \) and \( (x^2+3) \). By spotting these common factors, we can extract them to ease the simplification process.

Here's a quick guide to identifying common factors:
  • Look for repeated variables and constants in each term.
  • Focus on the highest power of each factor that appears in every term.
  • Underline or highlight these common elements for visual aid.
By spotting common factors, you set the stage for simplifying complex expressions effortlessly.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is the key to unlocking more advanced mathematical concepts. Once common factors are identified, we can combine like terms and reduce the complexity of the given expression.

Consider the factor extraction in the initial expression: \( (2x+1)^{-\frac{1}{2}}(x^2+3)^{-\frac{3}{2}}((x^2+3) - x(2x+1)) \).
This technique reduces the burden of tackling larger powers and messy coefficients.

Important steps:
  • Combine like terms within the brackets.
  • Ensure that addition and subtraction are correctly applied.
  • Continually check your work against the original expression.
By simplifying correctly, you conserve the integrity of the equation and pave the way for accurate solutions.
Techniques for Factoring
Factoring is a cornerstone technique in algebra. It breaks down complex expressions into more manageable parts.

Different factoring methods include:
  • Factor by grouping: This method involves grouping terms to facilitate extraction of common factors.
  • Difference of squares: Useful for expressions in the form \( a^2 - b^2 = (a-b)(a+b) \).
  • Factoring trinomials: Common for quadratic expressions, converting them into two binomial expressions.
In our example, we used grouping and extraction of common factors, simplifying \( (2x+1)^{-\frac{1}{2}}(x^2+3)^{-\frac{3}{2}} \). We then simplified the inner expression: \( (x^2+3) - x(2x+1) = -x^2 - x + 3 \).

Mastering these techniques will make algebraic problems more approachable and solvable.
Handling Negative Exponents
Negative exponents indicate division or reciprocals. Understanding how to manipulate them is essential for simplifying expressions. Here's a quick look at their properties:
  • \( a^{-n} = \frac{1}{a^n} \)
  • When multiplying, add the exponents: \( a^m \times a^n = a^{m+n} \)
  • When dividing, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \)
Applying these rules simplifies expressions effectively.

In our example, \( (2x+1)^{-\frac{1}{2}} \) and \( (x^2+3)^{-\frac{3}{2}} \) are negative exponents. By extracting these as common factors, it simplifies further steps.

Remember, changing negative exponents to positive ones by taking the reciprocal can often make the expression more intuitive and manageable.

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Most popular questions from this chapter

Area of a Dodecagon Part I A regular dodecagon is a polygon with 12 sides of equal length. See the figure. (a) The area \(A\) of a regular dodecagon is given by the formula \(A=12 r^{2} \tan \frac{\pi}{12},\) where \(r\) is the apothem, which is a line segment from the center of the polygon that is perpendicular to a side. Find the exact area of a regular dodecagon whose apothem is 10 inches. (b) The area \(A\) of a regular dodecagon is also given by the formula \(A=3 a^{2} \cot \frac{\pi}{12},\) where \(a\) is the length of a side of the polygon. Find the exact area of a regular dodecagon if the length of a side is 15 centimeters.

A beam of light with a wavelength of 589 nanometers traveling in air makes an angle of incidence of \(40^{\circ}\) on a slab of transparent material, and the refracted beam makes an angle of refraction of \(26^{\circ} .\) Find the index of refraction of the material. \({ }^{t}\)

Is the function \(f(x)=\frac{3 x}{5-x^{2}}\) even, odd, or neither?

Establish each identity. $$ \cos (\theta+k \pi)=(-1)^{k} \cos \theta, k \text { any integer } $$

Use the following discussion. The formula $$ D=24\left[1-\frac{\cos ^{-1}(\tan i \tan \theta)}{\pi}\right] $$ Approximate the number of hours of daylight in New York, New York \(\left(40^{\circ} 45^{\prime}\right.\) north latitude \()\), for the following dates: (a) Summer solstice \(\left(i=23.5^{\circ}\right)\) (b) Vernal equinox \(\left(i=0^{\circ}\right)\) (c) July \(4\left(i=22^{\circ} 48^{\prime}\right)\)
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