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A rational expression is simply a fraction where both the numerator and the denominator are polynomials. It extends the concept of fractions to algebraic expressions. Consider \[\frac{x+2}{(x+1)(x-1)}\]. Here, the numerator is \(x + 2\) and the denominator is the product of two polynomials, \(x+1\) and \(x-1\). This makes it a rational expression.

Rational expressions can have variables in both the numerator and the denominator, unlike regular fractions that only have numeric terms. Understanding how to manipulate these expressions is crucial in algebra, especially when simplifying them, finding common denominators, or performing operations like addition and subtraction.

To further simplify or decompose these expressions, we often use techniques like partial fraction decomposition, especially when solving calculus problems or integrating rational functions.

It's important to remember that rational expressions are undefined when the denominator equals zero. This concept leads to restrictions in the domain of these functions, often appearing in the form of vertical asymptotes in graphs.

Linear factors are expressions of the form \(ax + b\), where both \(a\) and \(b\) are constants. They represent lines when graphed, hence the name 'linear'.

In the partial fraction decomposition of a rational expression, identifying linear factors is crucial because they determine the form of the decomposition. For the expression \(\frac{x+2}{(x+1)(x-1)}\), the denominator is already factored into two distinct linear factors: \((x+1)\) and \((x-1)\).

Each linear factor corresponds to a separate fraction in the decomposition, for which constants \(A\), \(B\), etc., are solved:

• \(\frac{A}{x+1}\)
• \(\frac{B}{x-1}\)

Understanding linear factors is not only fundamental in partial fraction decomposition but also in other areas like solving quadratic equations, graphing linear equations, and analyzing polynomial functions.

The unique characteristics of each factor, such as their roots, influence how the rational expression behaves or is simplified.

A system of equations consists of two or more equations with the same set of variables. Solving a system means finding values for the variables that satisfy all equations simultaneously.

In partial fraction decomposition, we often set up a system of equations to determine the unknown constants in the decomposed fractions. Let's consider the equations from the example:

• \(A + B = 1\)
• \(B - A = 2\)

This system arises when we compare the coefficients from both sides of the expanded equation, derived after equating and combining terms.

We solve this system using substitution or elimination methods. For these equations, adding them eliminates \(A\), allowing us to solve for \(B\). Then, substituting back to find \(A\) gives us the complete solution.

This technique not only applies to partial fraction decomposition but is also widely used in fields like physics, economics, and engineering, wherever multiple conditions are imposed on a set of variables.