91Ó°ÊÓ

Factoring is a super important skill when dealing with algebraic expressions, especially in rational expressions. In algebra, factoring means breaking down a complex expression into simpler, multiplied components. This is much like breaking down a number into its prime factors.For example, consider the expression \(x^2 - 9\). This is a difference of squares that can be factored into:

• \((x-3)(x+3)\)

Another expression we might encounter is \(x^2 - 6x + 9\). Recognize this as a perfect square trinomial, factoring to:

• \((x-3)(x-3)\) or \((x-3)^2\)

When solving rational expressions, factoring helps identify the common denominator, which is crucial for addition or subtraction. Factoring simplifies the expressions as much as possible, preparing them for these operations. By identifying these factors, the process of finding a common denominator becomes more straightforward.

Finding a common denominator is the trick to adding or subtracting fractions, whether numerical or algebraic. It is similar to finding a common language that all fractions can "speak."In the original exercise, we have three denominators: \(x^2 - 9\), \(x^2 - 6x + 9\), and \(x + 3\). Factoring these gives:

• \((x-3)(x+3)\) for \(x^2 - 9\)
• \((x-3)^2\) for \(x^2 - 6x + 9\)
• \((x+3)\) is already factored

The common denominator becomes \((x-3)^2(x+3)\). Having this common denominator allows us to combine all terms into a single fraction.To achieve this, adjust each fraction by multiplying both the numerator and the denominator by the necessary terms to reach the common denominator. This way, fractions seamlessly come together under a unified denominator.

Once fractions share a common denominator, the next step is simplification. This involves combining the fractions and reducing them as much as possible.After rewriting each fraction with the common denominator \((x-3)^2(x+3)\), merge them to form:

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

Next comes simplifying the numerator. Expand each component and combine the like terms:

• \(3(x-3) = 3x - 9\)
• \(x(x+3) = x^2 + 3x\)
• \((x-3)^2 = x^2 - 6x + 9\)

Combining these terms gives:

• \(-6x\)

Ultimately, you end up with the fraction:

• \[\frac{-6x}{(x-3)^2(x+3)}\]

Check if further simplification is possible by identifying any common factors between the numerator and the denominator. In this case, since there are no common factors, this fraction is simplified as far as it can go. Simplicity is key in mathematics, and reducing expressions to their simplest form makes them much easier to understand.