Problem 55 - Algebra and Trigonometry

Question

Perform the addition or subtraction and simplify. $$ \frac{1}{x^{2}}+\frac{1}{x^{2}+x} $$

Step-by-Step Solution

Answer

\(\frac{2x + 1}{x^2(x+1)}\) is the simplified result.

Step 1: Identify the Operation

The exercise is to add two rational expressions: \(\frac{1}{x^{2}}\) and \(\frac{1}{x^{2}+x}\). To perform addition, we need to find a common denominator.

Step 2: Determine a Common Denominator

The denominators of the two fractions are \(x^2\) and \(x^2 + x\). The least common denominator (LCD) is the least common multiple of these two expressions, which is \(x^2(x+1)\).

Step 3: Rewrite Fractions with the Common Denominator

Rewrite each fraction so that they have the denominator \(x^2(x+1)\). - The first fraction \(\frac{1}{x^2}\) becomes \(\frac{x+1}{x^2(x+1)}\) after multiplying the numerator and the denominator by \(x+1\).- The second fraction \(\frac{1}{x^2+x}\) becomes \(\frac{x}{x^2(x+1)}\) after multiplying the numerator and the denominator by \(x\).

Step 4: Add the Fractions

With a common denominator, add the numerators: \(\frac{x+1}{x^2(x+1)} + \frac{x}{x^2(x+1)} = \frac{(x+1) + x}{x^2(x+1)} = \frac{2x + 1}{x^2(x+1)}.\)

Step 5: Simplify

The expression \(\frac{2x + 1}{x^2(x+1)}\) cannot be simplified further because \(2x + 1\) has no common factors with \(x^2(x+1)\).

Key Concepts

Common Denominator

When adding or subtracting fractions, having a common denominator is essential. A common denominator allows us to add or subtract the numerators directly. Here's how it works:

• Both fractions must be expressed over the same denominator.
• The denominator becomes the least common multiple of the individual denominators of the fractions.

By finding a common denominator, such as in our exercise with \(x^2\) and \(x^2+x\), we effectively create a common platform on which the numerators can be combined. This key step avoids any disruption from differing denominators and is crucial for accurately performing fraction operations.

Least Common Multiple

The least common multiple (LCM) of two expressions is the smallest expression that both denominators can divide into without leaving a remainder. Finding the LCM helps us get a common denominator for rational expressions.

• It involves factoring each expression fully.
• Then combine the factors, taking the highest power of each factor present in any of the expressions.

For our exercise, the denominator \(x^2\) factors into \(x \times x\), and \(x^2 + x\) can be factored out as \(x(x+1)\). The LCM, therefore, includes \(x^2(x+1)\) because it covers all factors from both expressions. This denominator will enable a straightforward addition of the two fractions.

Adding Fractions

Once fractions are rewritten to have a common denominator, we can proceed with addition. In doing so:

• Align the fractions with the common denominator.
• Add the numerators together, while keeping the common denominator constant.

In our example, after modifying the fractions to \(\frac{x+1}{x^2(x+1)}\) and \(\frac{x}{x^2(x+1)}\), we simply add these numerators: \(x+1\) and \(x\), resulting in \(2x + 1\). This sum becomes the numerator of the resulting fraction over the common denominator. Remember, the denominator does not change during this process.

Simplifying Expressions

After combining fractions, simplification is the next step. The primary aim is to express the result in its simplest form. Simplification can involve:

• Canceling any common factors between the numerator and denominator.
• Ensuring the expression is fully reduced so there are no further common factors.

In our context, the final expression of \(\frac{2x + 1}{x^2(x+1)}\) is already in its simplest form since \(2x + 1\) shares no common factors with \(x^2(x+1)\). If common factors existed, they would be canceled out. Simplification helps in achieving a more readable result and ensures no unnecessary complexity is left in the expression.