Problem 32 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{5 x}{(x-2)^{2}}-\frac{3}{x-2}\)

Step-by-Step Solution

Answer

\( \frac{2(x + 3)}{(x-2)^2} \)

Step 1: Identify the Least Common Denominator

The expression contains two fractions. The denominators are \( (x-2)^2 \) and \( x-2 \). To perform the operation, we first need to find the least common denominator (LCD), which is \( (x-2)^2 \).

Step 2: Rewrite Each Fraction with the Common Denominator

For the first fraction, \( \frac{5x}{(x-2)^2} \), the denominator is already the LCD, so it stays the same. For the second fraction, \( \frac{3}{x-2} \), we multiply both the numerator and the denominator by \( (x-2) \) to get a common denominator: \[\frac{3(x-2)}{(x-2)^2} = \frac{3x - 6}{(x-2)^2}\]

Step 3: Combine the Fractions

Since both fractions now have the same denominator, we can subtract them: \[\frac{5x}{(x-2)^2} - \frac{3x - 6}{(x-2)^2} = \frac{5x - (3x - 6)}{(x-2)^2}\]

Step 4: Simplify the Numerator

Distribute the negative sign and simplify the expression in the numerator: \[5x - 3x + 6 = 2x + 6\]

Step 5: Write the Simplified Fraction

Now place the simplified numerator over the common denominator: \[\frac{2x + 6}{(x-2)^2}\]

Step 6: Simplify Further If Possible

We check if \( 2x + 6 \) can be factored further. Factoring gives \( 2(x + 3) \), so the expression becomes: \[\frac{2(x + 3)}{(x-2)^2}\]. However, it cannot be simplified further because \( x + 3 \) and \( (x-2)^2 \) have no common factors.

Key Concepts

Least Common Denominator in Algebraic Fractions

When working with algebraic fractions, such as the expression \( \frac{5x}{(x-2)^2} - \frac{3}{x-2} \), identifying the least common denominator (LCD) is crucial. The LCD is the smallest expression that both denominators can divide without leaving a remainder. This is similar to finding the least common multiple in arithmetic but applied to expressions or polynomials.

For our example, the denominators are \((x-2)^2\) and \(x-2\). To find the LCD, we look for the expression that each denominator can fit into evenly. Here, \( (x-2)^2 \) serves as the LCD because it already incorporates \(x-2\). This ensures that every term has a common baseline or denominator, necessary for combining the fractions.

Identifying the correct LCD allows us to manipulate each fraction so that their denominators match. This step lays the foundation for subtracting or adding fractions smoothly.

Simplifying Expressions in Algebra

Simplifying expressions involves reducing an algebraic expression to its simplest form. This process often involves rewriting fractions using the least common denominator and then combining and reducing them.

In our example, after adjusting the second fraction \( \frac{3}{x-2} \) to have the common denominator \( (x-2)^2 \), we multiply both its numerator and denominator by \(x-2\) to become \(\frac{3(x-2)}{(x-2)^2} = \frac{3x - 6}{(x-2)^2}\).

• Combining fractions involves aligning them with the same denominator.
• Once aligned, we perform operations on the numerators.

Next, we subtract these adjusted fractions. With the common denominator in place, the subtraction itself happens only on the numerators: \( \frac{5x - (3x-6)}{(x-2)^2} \).

Simplifying further requires distributing any negative signs and combining like terms: \(5x - 3x + 6\), simplifying to \(2x + 6\). Finally, if possible, factor the expression further to check for additional simplification. Here, factoring \(2x + 6\) yields \(2(x + 3)\), which brings us to the simplest form of the original expression when placed over the LCD \((x-2)^2\).

Subtracting Fractions with Algebraic Expressions

Subtracting fractions in the realm of algebra follows a familiar path to subtracting numerical fractions, but with an algebraic twist. It's essential to have a common denominator before performing any subtraction.

In algebra, subtracting fractions like \( \frac{5x}{(x-2)^2} - \frac{3}{x-2} \) requires careful manipulation. After establishing a common denominator \((x-2)^2\), direct subtraction is achieved by focusing on the numerators: \(5x - (3x - 6)\).

• First, distribute the negative sign across the terms of the second numerator.
• Combine like terms from both numerators.
• Place the new numerator over the common denominator.

The careful management of signs and terms ensures that the subtraction process correctly simplifies to \(\frac{2x + 6}{(x-2)^2}\). This approach ensures mathematical precision and clarity in transforming complex expressions into simpler, more manageable forms.