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When simplifying fractions, finding a common denominator is crucial. It ensures both fractions can be combined seamlessly. In algebra, variables and expressions form our denominators. Let's use the expression \( \frac{x}{x^{2}-9} + \frac{5x}{x-3} \).

We first determine a denominator that both fractions share. The denominator \( x^2 - 9 \) is a difference of squares. It factors into \((x-3)(x+3)\).

Assess the second denominator, \(x-3\). Here, it already appears in the first denominator's factored form, leaving us to include the additional factor \((x+3)\) to match. That's how a common denominator of \((x-3)(x+3)\) is determined.

When fractions share a common denominator, algebraic operations like addition, subtraction, and simplification become possible. This step is foundational in successful algebraic simplification.

Factoring plays a big role in algebraic simplification. It allows us to break down complex expressions into simpler parts. Let’s consider the expression with denominators \(x^2 - 9\) and \(x-3\). The first task is factoring \(x^2 - 9\), a difference of squares.

A difference of squares is an expression like \(a^2 - b^2\), which factors into \((a+b)(a-b)\).

• Here, \(x^2 - 9\) represents \(a^2 - b^2\) with \(a = x\) and \(b = 3\).
• So, it becomes \((x+3)(x-3)\).

With the numerator \(5x^2 + 16x\) in the expression \(\frac{x(5x + 16)}{(x-3)(x+3)}\), factoring requires finding common factors.

• We can factor out \(x\), giving \(x(5x + 16)\).

Factoring simplifies expressions and is a key step in uncovering potential simplifications in both numerators and denominators.

Once each fraction has a common denominator, the path to combining them opens up. Consider \(\frac{x}{(x-3)(x+3)} + \frac{5x(x+3)}{(x-3)(x+3)}\). Both share the common denominator \((x-3)(x+3)\).

Combine the fractions by adding or subtracting the numerators.

• Add the numerators to create \(x + 5x(x+3)\).
• Simplifying this involves expanding \(5x(x+3)\) to \(5x^2 + 15x\).
• Combine like terms: \(x + 5x^2 + 15x\) becomes \(5x^2 + 16x\).

Combining fractions results in a single simplified fraction. This method is essential when solving algebraic expressions with fractions, as it reduces complexity and paves the way for further simplification or solution.