Problem 32 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{7}{x+2}+\frac{3 x}{(x+2)^{2}}-\frac{5}{x}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-2x^2 - 13x - 20}{x(x+2)^2}\).

Step 1: Identify the Least Common Denominator (LCD)

To simplify the expression, we need to identify a common denominator for all the fractions. Here, the denominators are \(x+2\), \((x+2)^2\), and \(x\). The least common denominator is therefore \((x)(x+2)^2\) as it incorporates the highest power of \(x+2\) and includes \(x\).

Step 2: Rewrite Each Term with the LCD

Convert each fraction to have the least common denominator \((x)(x+2)^2\). - The term \(\frac{7}{x+2}\) becomes \(\frac{7x}{(x)(x+2)^2}\) by multiplying the numerator and denominator by \(x(x+2)\).- The term \(\frac{3x}{(x+2)^2}\) becomes \(\frac{3x^2}{(x)(x+2)^2}\) by multiplying the numerator and denominator by \(x\).- The term \(\frac{5}{x}\) becomes \(\frac{5(x+2)^2}{(x)(x+2)^2}\) by multiplying the numerator and denominator by \((x+2)^2\).

Step 3: Combine the Fractions

Now that all fractions have the same denominator, they can be combined into a single fraction:\[\frac{7x}{(x)(x+2)^2} + \frac{3x^2}{(x)(x+2)^2} - \frac{5(x+2)^2}{(x)(x+2)^2} = \frac{7x + 3x^2 - 5(x+2)^2}{(x)(x+2)^2}\]

Step 4: Simplify the Numerator

Simplify the expression in the numerator:1. Expand \(-5(x+2)^2\) to get \(-5(x^2 + 4x + 4) = -5x^2 - 20x - 20\).2. Combine the terms \(3x^2 + 7x - 5x^2 - 20x - 20\). - This results in \(-2x^2 - 13x - 20\).

Step 5: Write the Simplified Expression

Finally, write the simplified expression:\[\frac{-2x^2 - 13x - 20}{(x)(x+2)^2}\]

Step 6: Verify the Simplification Process

Check each step for arithmetic errors and confirm that the simplification is correct by reviewing calculations and logical flow.

Key Concepts

Simplifying Rational Expressions

Rational expressions are similar to fractions but involve polynomials in the numerator and/or the denominator. Simplifying rational expressions requires a good understanding of polynomial algebra. It's about reducing the expression to its simplest form without changing its value. For example, in our exercise, we encounter the expression \(\frac{7}{x+2} + \frac{3x}{(x+2)^2} - \frac{5}{x}\). To simplify it, we need to:

• Find the least common denominator (LCD), which is a crucial step in combining fractions.
• Rewrite each term so that all terms have this common denominator.
• Combine and simplify the terms into a single simplified expression.

Simplification involves balancing the need to maintain equivalent expressions while reducing complexity.

Combining Fractions

When adding or subtracting fractions, it's essential to have a shared denominator. This often involves finding the Least Common Denominator (LCD) to ensure each fraction is comparable. In the given expression, the denominators \(x+2\), \((x+2)^2\), and \(x\) need a common base. The LCD in this instance is \((x)(x+2)^2\). Once each fraction is rewritten to have this shared base, they can be added or subtracted directly. Here's how you rewrite each term:

• Multiply both the numerator and the denominator of each term by the necessary factors to achieve the LCD.
• Combine the fractions into one single fraction with the LCD as the new denominator.

This unified structure allows for easier manipulation and simplification of the algebraic expression.

Polynomial Expressions

A polynomial expression includes variables, coefficients, and exponents. Simplifying such expressions involves combining like terms, factoring, and applying algebraic operations efficiently.In our example, once we combine the fractions, we must simplify the expression by focusing on the numerator. This requires:

• Expanding and combining polynomial terms.
• Carefully managing positive and negative signs to avoid errors.

Our solution's numerator becomes the expression \(7x + 3x^2 - 5(x+2)^2\). Expanding and simplifying this requires calculating \(-5(x^2 + 4x + 4) = -5x^2 - 20x - 20\) and then combining it with other terms like \(3x^2 + 7x\). This leads to a simplified polynomial in the numerator \(-2x^2 - 13x - 20\).

Fraction Operations

Fraction operations involve adding, subtracting, multiplying, or dividing fractions. Each operation has specific rules, primarily centered around maintaining a common denominator for addition and subtraction. In this problem, we focused on addition and subtraction of fractions:

• Identifying and utilizing the LCD to align fractions.
• Restructuring fractional terms using symbolic multiplication to make addition/subtraction feasible.

The critical aspect of fraction operations is converting each fraction appropriately and then simplifying the equation. With a shared denominator, operations become straightforward, and combining terms in the numerator results in a reductive process, arriving at a simplified expression. Finally, check your work to ensure all calculations are correct and the logic is sound. This ensures the rational expression is neatly reduced to its simplest form.