Pure: Partial Fractions

A. Rational Function Definition: A rational function is a function that can be written as the ratio of two polynomials, P(x) and Q(x), where Q(x) is not identically zero. Explanation: Think of a rational function as a fraction where both the numerator and the denominator are polynomials. For example, (x+1)/(x²-4) is a rational function. The domain of such a function includes all real numbers except those values of x that make the denominator Q(x) equal to zero, as division by zero is undefined. Think of it as: Just as a rational number is a fraction of two integers, a rational function is a 'fraction' of two polynomials. It's like having a complex recipe where both the ingredients list and the cooking instructions are themselves formulas. Real-world application: Rational functions are used to model various real-world phenomena, such as the concentration of a chemical in a solution over time, the cost per item in a production process, or the behavior of electrical circuits. They help describe situations where quantities change in relation to each other in a non-linear way. B.