Problem 28 - Algebra 1: Concepts and Skills

Question

FACTORING AFTER ADDING OR SUBTRACTING. Simplify the expression. $$ \frac{2 x}{x^{2}+5 x}-\frac{x}{x^{2}+5 x} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{x + 5}\)

Step 1: Identifying Denominator

The denominator of both fractions in this case is \(x^{2} + 5x\), which makes the calculation straightforward since the denominators are already equalized.

Step 2: Performing Subtraction

With common denominators, subtraction of the fractions reduces to subtracting the numerators, while keeping the common denominator the same. This results in: \(\frac{2x - x}{x^{2} + 5x}= \frac{x}{x^{2} + 5x}\)

Step 3: Simplification

The expression \(\frac{x}{x^{2} + 5x}\) can be further simplified as \(\frac{1}{x + 5}\) after factoring out \(x\) in the denominator.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in algebra, particularly when dealing with complex expressions. To simplify a fraction, you look for common factors in the numerator (the top number) and the denominator (the bottom number) that can be divided out.

For instance, consider \frac{x}{x^2 + 5x}\. Here, the denominator can be factored to \frac{x}{x(x + 5)}\. By cancelling out the common \(x\) from the numerator and the denominator, the simplified form becomes \frac{1}{x + 5}\.

This process not only makes the expression cleaner, but also easier to work with in further operations such as addition, subtraction, multiplication, or division.

Subtracting Fractions

Subtracting fractions requires a common denominator, which allows you to directly subtract the numerators (the top part of the fractions). The fraction in our exercise, \(\frac{2x}{x^2+5x}-\frac{x}{x^2+5x}\), demonstrates this well. Because the denominators are identical, we can combine the terms by subtracting the numerators: \(2x - x\), which simplifies to \(x\).

The result, then, is the simplified fraction \(\frac{x}{x^2+5x}\), where the denominator stays as is, acting as the common ground for the operation. Remember, without a common denominator, additional steps would be required to first find one before proceeding with the subtraction.

Common Denominators

A common denominator is needed when performing operations such as addition or subtraction with fractions. The denominator represents the total number of equal parts in the whole, and it's essential that both (or all) fractions being combined refer to the same 'whole' when calculating.

In this exercise, the common denominator is already provided as \(x^2+5x\). Whenever you have different denominators, you would need to find a common multiple that both denominators can divide into evenly before you can proceed with the operation. But here, with a shared base, the focus moves to handling the numerators, simplifying the process.

Algebraic Factorization

Algebraic factorization is the process of breaking down expressions into their simplest form or into a product of factors. When dealing with fractions, factorization can help significantly in simplification.

In our exercise, for instance, the denominator \(x^2 + 5x\) can be factored to reveal a common \(x\) factor, becoming \(x(x + 5)\). It's a key step that enables reducing the fraction to its simplest form. Not only does factorization assist in the simplification process, but it can also be a powerful tool for solving equations, finding roots of polynomials, and simplifying complex algebraic expressions.