Problem 51 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified result is \( \frac{3x + 2}{(x+1)^{2}} \).

Step 1: Identify the Common Denominator

The fractions given are \( \frac{x}{(x+1)^{2}} \) and \( \frac{2}{x+1} \). To add these fractions, we need a common denominator. The least common denominator (LCD) is \((x+1)^{2}\).

Step 2: Rewrite the Second Fraction

To combine the fractions, rewrite \( \frac{2}{x+1} \) with the LCD. Multiply the numerator and denominator by \(x+1\): \( \frac{2}{x+1} = \frac{2(x+1)}{(x+1)^2} = \frac{2x + 2}{(x+1)^2} \).

Step 3: Combine the Fractions

Now that both fractions have the common denominator \((x+1)^2\), add their numerators: \( \frac{x}{(x+1)^{2}} + \frac{2x + 2}{(x+1)^{2}} = \frac{x + 2x + 2}{(x+1)^{2}} \).

Step 4: Simplify the Numerator

Simplify the numerator \(x + 2x + 2\): \(3x + 2\). Thus, the expression becomes \( \frac{3x + 2}{(x+1)^{2}} \).

Step 5: Final Result

The simplified form of the expression is \( \frac{3x + 2}{(x+1)^{2}} \). No further simplification is possible.

Key Concepts

Common Denominator

When dealing with fractions, especially those involving variables, finding a common denominator is essential. This allows us to perform addition or subtraction of these fractions.

• The common denominator is a shared baseline that both fractions can abide by.
• To determine this, look at the denominators of all fractions involved.
• The goal is to find the least common denominator (LCD), which is the smallest expression that all original denominators can divide into without leaving a remainder.

For the fractions \( \frac{x}{(x+1)^{2}} \) and \( \frac{2}{x+1} \), the LCD is \( (x+1)^{2} \). This is because the second fraction can be modified by this denominator without changing its value, enabling a straightforward addition of both fractions.

Addition and Subtraction of Fractions

Once a common denominator is established, the operation of addition or subtraction becomes much simpler.

• Each fraction must be expressed with the common denominator before performing any arithmetic operations on them.
• This may involve multiplying the numerator and denominator of a fraction by the same expression, ensuring that you do not change the fraction's value.

In our example, \( \frac{2}{x+1} \, \) had to be rewritten to \( \frac{2(x+1)}{(x+1)^2} = \frac{2x + 2}{(x+1)^{2}} \). Now that both fractions have the common denominator of \( (x+1)^2 \), they can be easily combined:\[ \frac{x}{(x+1)^{2}} + \frac{2x + 2}{(x+1)^{2}} = \frac{x + 2x + 2}{(x+1)^{2}} \]

Simplification of Expressions

After performing the operations, it's important to simplify the resulting expression.

• Simplification involves combining like terms and reducing the expression to its simplest form.
• The aim is to make the expression as compact as possible without changing its value.

In the example provided, the combined numerator \( x + 2x + 2 \) simplifies to \( 3x + 2 \). Thus, the overall expression is:\[ \frac{3x + 2}{(x+1)^{2}} \]No further simplification is possible for this fraction as there are no common factors between the numerator and the denominator that can be canceled out. Simplifying expressions often makes them easier to understand and use in further algebraic manipulations.