Problem 27
Question
Perform the indicated operations and simplify. \(\frac{12}{x^{2}+2 x}+\frac{4}{x}+\frac{2}{x+2}\)
Step-by-Step Solution
Verified Answer
\(\frac{2(3x + 10)}{x(x+2)}\)
1Step 1: Simplify the First Fraction
The first fraction is \(\frac{12}{x^2 + 2x}\). Factor the denominator: \(x^2 + 2x = x(x + 2)\). So, the expression becomes \(\frac{12}{x(x + 2)}\).
2Step 2: Express the Second and Third Fractions with Common Denominator
The second fraction is \(\frac{4}{x}\). To combine it with the first fraction, multiply its numerator and denominator by \((x+2)\) to get \(\frac{4(x+2)}{x(x+2)}\). The third fraction is \(\frac{2}{x+2}\). Multiply its numerator and denominator by \(x\) to get \(\frac{2x}{x(x+2)}\).
3Step 3: Combine the Fractions
Now that all fractions have the common denominator \(x(x+2)\), combine them into one fraction: \(\frac{12 + 4(x+2) + 2x}{x(x+2)}\).
4Step 4: Simplify the Numerator
Expand and simplify the numerator: \(12 + 4(x+2) + 2x = 12 + 4x + 8 + 2x = 4x + 2x + 12 + 8 = 6x + 20\).
5Step 5: Rewrite the Expression
The combined fraction is \(\frac{6x + 20}{x(x+2)}\). Further simplify by factoring the numerator: \(6x + 20 = 2(3x + 10)\).
6Step 6: Final Simplified Form
The final simplified expression is \(\frac{2(3x + 10)}{x(x+2)}\).
Key Concepts
Fraction SimplificationLeast Common DenominatorFactoring Polynomials
Fraction Simplification
Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. This means eliminating or factoring out shared numbers or expressions to make calculations easier.
In our example, the first fraction is given as \(\frac{12}{x^2 + 2x}\). Simplification involves factoring the denominator \(x^2 + 2x\), which results in \(x(x + 2)\). Therefore, the fraction becomes \(\frac{12}{x(x + 2)}\).
Simplification is crucial because it makes it easier to combine this fraction with other fractions by aligning them under a common denominator.
In our example, the first fraction is given as \(\frac{12}{x^2 + 2x}\). Simplification involves factoring the denominator \(x^2 + 2x\), which results in \(x(x + 2)\). Therefore, the fraction becomes \(\frac{12}{x(x + 2)}\).
Simplification is crucial because it makes it easier to combine this fraction with other fractions by aligning them under a common denominator.
Least Common Denominator
When working with multiple fractions, especially in addition and subtraction, it's essential to express them with the same denominator. This common ground is known as the least common denominator (LCD).
The LCD is derived from the smallest number or expression that is a multiple of all denominators involved. In our case, we have the fractions \(\frac{4}{x}\) and \(\frac{2}{x+2}\). To combine these with \(\frac{12}{x(x+2)}\), each fraction must be expressed with the denominator \(x(x+2)\), which serves as our LCD.
The LCD is derived from the smallest number or expression that is a multiple of all denominators involved. In our case, we have the fractions \(\frac{4}{x}\) and \(\frac{2}{x+2}\). To combine these with \(\frac{12}{x(x+2)}\), each fraction must be expressed with the denominator \(x(x+2)\), which serves as our LCD.
- For \(\frac{4}{x}\), multiply both the numerator and the denominator by \(x+2\) to make it \(\frac{4(x+2)}{x(x+2)}\).
- For \(\frac{2}{x+2}\), multiply both the numerator and the denominator by \(x\) to achieve \(\frac{2x}{x(x+2)}\).
Factoring Polynomials
Factoring polynomials is a powerful technique used in algebra to rewrite polynomial expressions as the product of simpler polynomials. This is often the initial step in solving equations or simplifying expressions because it reveals hidden commonalities.
Applying factoring to our problem helps reveal common denominators and simplifies addition and subtraction operations. For instance, the expression \(x^2 + 2x\) can be factored into \(x(x+2)\), which is straightforward yet crucial for simplification.
Additionally, when combining all the fractions into one, we arrived at the fraction \(\frac{6x + 20}{x(x+2)}\). Factoring the numerator \(6x + 20\) as \(2(3x + 10)\) enables further simplification, ensuring the expression is as reduced as possible: \(\frac{2(3x + 10)}{x(x+2)}\).
Understanding how to factor polynomials effectively can greatly aid in reducing more complex algebraic expressions to simpler, more manageable forms.
Applying factoring to our problem helps reveal common denominators and simplifies addition and subtraction operations. For instance, the expression \(x^2 + 2x\) can be factored into \(x(x+2)\), which is straightforward yet crucial for simplification.
Additionally, when combining all the fractions into one, we arrived at the fraction \(\frac{6x + 20}{x(x+2)}\). Factoring the numerator \(6x + 20\) as \(2(3x + 10)\) enables further simplification, ensuring the expression is as reduced as possible: \(\frac{2(3x + 10)}{x(x+2)}\).
Understanding how to factor polynomials effectively can greatly aid in reducing more complex algebraic expressions to simpler, more manageable forms.
Other exercises in this chapter
Problem 27
In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(g(x)=\left\lceil\frac{x}{2}\right\rceil\)
View solution Problem 27
Tell whether each of the following is true or false. (a) \(-3-17\) (c) \(-3
View solution Problem 28
Find the exact values in Problems 27-31. Hint: Half-angle identities may be helpful. $$ \sin ^{2} \frac{\pi}{6} $$
View solution Problem 28
In Problems 23-28, find the slope of the line containing the given two points. \((-6,0)\) and \((0,6)\)
View solution