Problem 37 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{2 x}{x-7}-\frac{x}{x-2}\)

Step-by-Step Solution

Answer

\(\frac{x^2 + 3x}{(x-7)(x-2)}\)

Step 1: Identify the Least Common Denominator (LCD)

To subtract these fractions, we need a common denominator. Both denominators \(x-7\) and \(x-2\) are linear polynomials, so their Least Common Denominator is the product of both: \((x-7)(x-2)\).

Step 2: Rewrite Each Fraction

Convert each fraction into an equivalent fraction with the LCD \((x-7)(x-2)\). For \(\frac{2x}{x-7}\), multiply the numerator and the denominator by \(x-2\), resulting in \(\frac{2x(x-2)}{(x-7)(x-2)}\). For \(\frac{x}{x-2}\), multiply the numerator and the denominator by \(x-7\), resulting in \(\frac{x(x-7)}{(x-7)(x-2)}\).

Step 3: Combine the Fractions

Now that both fractions have the same denominator, subtract the second fraction from the first: \[\frac{2x(x-2)}{(x-7)(x-2)} - \frac{x(x-7)}{(x-7)(x-2)} = \frac{2x(x-2) - x(x-7)}{(x-7)(x-2)}\]

Step 4: Simplify the Numerator

Distribute and combine like terms in the numerator: \[2x(x-2) = 2x^2 - 4x \quad \text{and} \quad x(x-7) = x^2 - 7x\]Subtract the second expression from the first: \[2x^2 - 4x - (x^2 - 7x) = 2x^2 - 4x - x^2 + 7x = x^2 + 3x\]

Step 5: Write the Final Simplified Expression

Place the simplified numerator \(x^2 + 3x\) over the common denominator: \[\frac{x^2 + 3x}{(x-7)(x-2)}\]

Key Concepts

Least Common Denominator

When dealing with fractions, especially algebraic ones, it's essential to have a common denominator before performing operations such as addition or subtraction. In algebra, finding this common denominator is known as identifying the Least Common Denominator (LCD). The LCD is the smallest expression that can contain all the denominators of the fractions involved.

• In our example with fractions \(\frac{2x}{x-7}\) and \(\frac{x}{x-2}\), both denominators are linear polynomials.
• The LCD, therefore, is the product of these two polynomials, resulting in \((x-7)(x-2)\).

By obtaining the LCD, you ensure that the operation you're performing applies to expressions that share a common base, which simplifies the subtraction or addition process. In mathematical terms, the LCD lets you convert unlike fractions into like fractions, easing the calculation process.

Subtraction of Fractions

Once you've found the Least Common Denominator, the next step is to rewrite each fraction so that they share this common denominator. Subtraction of fractions becomes straightforward when they have the same denominator:

• First, express both fractions using the LCD as their denominator.
• For the fraction \(\frac{2x}{x-7}\), multiply both the numerator and the denominator by \(x-2\).
• Similarly, for \(\frac{x}{x-2}\), multiply both the numerator and the denominator by \(x-7\).

This conversion process allows you to align the fractions, making it easier to subtract one from the other. The fractions are then expressed as \(\frac{2x(x-2)}{(x-7)(x-2)}\) and \(\frac{x(x-7)}{(x-7)(x-2)}\).
Place these expressions over the common denominator and subtract the second fraction from the first, focusing solely on subtracting the numerators. The denominator remains \((x-7)(x-2)\) throughout the process.

Polynomial Simplification

After setting up the subtraction of fractions with a common denominator, the next key step is polynomial simplification. The focus is on the numerators, which may need expanding and then combining like terms. It helps achieve the simplest form of the expression.
To simplify the numerators, carry out these steps:

• Distribute the terms inside the brackets to remove them, turning \(2x(x-2)\) into \(2x^2 - 4x\) and \(x(x-7)\) into \(x^2 - 7x\).
• Perform the subtraction: \((2x^2 - 4x) - (x^2 - 7x)\).
• The result simplifies to \(x^2 + 3x\), achieved by combining like terms.

Afterward, place the simplified polynomial back over the original common denominator \((x-7)(x-2)\). This step ensures that the expression is reduced to its most concise form, highlighting any patterns or factors that simplify further exploration or integration into other mathematical problems.