Problem 10 - Elementary Algebra

Question

For Problems 1-40, perform the indicated operations and express answers in simplest form. $$ \frac{3 x}{x^{2}+2 x}+\frac{4}{5 x} $$

Step-by-Step Solution

Answer

\( \frac{19x + 8}{5x(x + 2)} \)

Step 1: Find a common denominator

To add the two fractions \( \frac{3x}{x^2 + 2x} \) and \( \frac{4}{5x} \), we need a common denominator. The denominators are \( x^2 + 2x \) and \( 5x \). The least common denominator (LCD) is \( 5x(x + 2) \).

Step 2: Rewrite each fraction with the common denominator

Express each fraction with the common denominator \( 5x(x + 2) \).For \( \frac{3x}{x^2 + 2x} = \frac{3x}{x(x+2)} \), multiply the numerator and the denominator by \( 5 \) to get: \[\frac{15x}{5x(x+2)}\]For \( \frac{4}{5x} \), multiply the numerator and the denominator by \( x + 2 \) to get: \[\frac{4(x+2)}{5x(x+2)}\]

Step 3: Add the fractions

Now that the fractions have the same denominator, add the numerators together:\[\frac{15x + 4(x+2)}{5x(x+2)}\]

Step 4: Simplify the numerator

Expand the numerator of the expression:\[15x + 4(x+2) = 15x + 4x + 8 = 19x + 8\]So, the expression becomes:\[\frac{19x + 8}{5x(x + 2)}\]

Step 5: Verify simplification

Check if the numerator \( 19x + 8 \) has any common factors with the denominator that can be further simplified. There are no common factors, so the expression is already in its simplest form.

Key Concepts

Common Denominator

When dealing with fractions, the key to adding them is to have a common denominator. A common denominator is a shared multiple of the denominators of two or more fractions. This allows you to easily combine the fractions into a single expression.

In our exercise, we worked with the fractions \( \frac{3x}{x^2 + 2x} \) and \( \frac{4}{5x} \). The denominators here are different, thus, finding a common denominator is necessary. A good approach is to determine the least common denominator (LCD). The LCD for our fractions is the smallest expression that both denominators can divide into without leaving a remainder. Here, the LCD is \( 5x(x+2) \). This ensures that the addition can be performed seamlessly once both fractions are rewritten to share this common denominator.

• Identify each denominator: \( x^2 + 2x \) can be factored to \( x(x+2) \), and the second is \( 5x \).
• The common denominator includes all unique factors: thus \( 5x(x+2) \).

Simplest Form

Expressing an answer in its simplest form means reducing it to the most basic version that contains no common factors between the numerator and denominator, other than one. Once we had our common denominator, the next steps involved combining and simplifying the fractions.

After rewriting each fraction to have the common denominator, we combined them: \[\frac{15x}{5x(x+2)} + \frac{4(x+2)}{5x(x+2)} = \frac{15x + 4(x+2)}{5x(x+2)}\] First, the numerator was expanded to \( 15x + 4x + 8 \), simplifying to \( 19x + 8 \). Next, we checked for common factors that could be cancelled out with those in the denominator \( 5x(x+2) \). Since there were none, the expression was already simplified.

• Expand and simplify the numerator.
• Check for common factors with the denominator.
• Ensure no further simplification is possible.

Polynomial Expressions

Polynomial expressions are mathematical phrases involving sums of powers in one or more variables multiplied by coefficients. In our problem, both fractions had polynomial denominators. The original denominators were \( x^2 + 2x \) (a quadratic polynomial) and \( 5x \) (a linear polynomial).

Understanding polynomial expressions is crucial as they often require factorization, especially when finding common denominators or simplifying expressions. For example, \( x^2 + 2x \) can be factored to \( x(x+2) \), which then helps us recognize the common factors quickly when handling the fractions.

• Recognize different types of polynomials: linear (degree 1), quadratic (degree 2), etc.
• Use factorization to simplify polynomials and identify shared factors.
• Understanding polynomials aids in manipulating and simplifying rational expressions.