91Ó°ÊÓ

Quadratic factorization is a fundamental process in algebra that involves breaking down a quadratic expression into simpler, multiplying expressions, called factors. This process is essential for solving equations and simplifying expressions. To factor a quadratic like \( x^2 - x - 6 \), we look for two numbers that multiply to give the constant term (in this case, \(-6\)) and add to give the linear coefficient (\(-1\)). After some trials, we find these numbers to be \(-3\) and \(+2\). This means we can write the quadratic as

• \((x - 3)(x + 2)\),

which are its factors. This transformation is crucial as it helps us later in finding a common denominator for rational expressions.

When working with multiple fractions that involve polynomials, having a common denominator is key to adding or subtracting them. A common denominator is a shared multiple of the denominators of all the fractions in the expression. In this exercise, we have fractions with denominators \((x-3)(x+2)\), \(x+2\), and \(x-3\). The common denominator involves combining the unique factors from each of these:

• \((x-3)(x+2)\).

Writing each fraction with this common base allows for easier and direct manipulation, such as combining numerators. This step converts each fraction into an equivalent fraction with the same denominator, setting up the fractions for straightforward addition or subtraction without worrying about unequal denominators.

Simplifying fractions is the final step in making rational expressions as concise as possible. Once we've rewritten each fraction with a common denominator, as in our original problem, simplifying involves merging the numerators into a single expression. This means:

• Performing all subtraction and addition operations within the numerators.

For our expression, this results in a numerator of \(-2x - 1\). The fraction then becomes \( \frac{-2x - 1}{(x-3)(x+2)} \). If there are common factors in the numerator and denominator, they can be canceled; however, in this case, there are none. Therefore, this fraction is the simplest form we can achieve.