rationalization

Examples of rationalization in the following topics:

• Zeroes of Polynomial Functions With Rational Coefficients

  • A real number that is not rational is called irrational.
  • The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers.
  • In mathematics, the adjective rational often means that the underlying field considered is the field Q of rational numbers.
  • Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a "polynomial over the rationals".
  • However, rational function does not mean the underlying field is the rational numbers, and a rational algebraic curve is not an algebraic curve with rational coefficients.

• Introduction to Rational Functions

  • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
  • Note that every polynomial function is a rational function with $Q(x) = 1$.
  • For a simple example, consider the rational function $y = 1/x$.
  • Factorizing the numerator and denominator of rational function helps to identify singularities of algebraic rational function.
  • Graph of a rational function with equation $\frac{(x^2 - 3x -2)}{(x^2 - 4)}$.

• Rational Decision Making

  • Rational decision making is a multi-step process, from problem identification through solution, for making logically sound decisions.
  • Rational decision making is a multi-step process for making choices between alternatives.
  • The process of rational decision making favors logic, objectivity, and analysis over subjectivity and insight.
  • The word "rational" in this context does not mean sane or clear-headed as it does in the colloquial sense.
  • The idea of rational choice is easy to see in economic theory.

• Problems with the Rational Decision-Making Model

  • Critics of rational choice theory—or the rational model of decision-making—claim that this model makes unrealistic and over-simplified assumptions.
  • Their objections to the rational model include:
  • The more complex a decision, the greater the limits are to making completely rational choices.
  • The theory of bounded rationality holds that an individual's rationality is limited by the information they have, the cognitive limitations of their minds, and the finite amount of time they have to make a decision.
  • Bounded rationality shares the view that decision-making is a fully rational process; however, it adds the condition that people act on the basis of limited information.

• Theory of Utility

  • The theory of utility is based on the assumption of that individuals are rational.
  • Rationality has a different meaning in economics than it does in common parlance.
  • It is important to emphasize how rationality relates to a person's individual preferences.
  • Based on their preferences, both made the economically rational choice.
  • The notion of rationality is therefore central to any understanding of microeconomics.

• Rational-Legal Authority

  • Rational-legal authority is a form of leadership in which authority is largely tied to legal rationality, legal legitimacy, and bureaucracy.
  • In rational-legal authority, power is passed on according to a set of rules.
  • Rational-legal authority is a form of leadership in which the authority of an organization or a ruling regime is largely tied to legal rationality, legal legitimacy, and bureaucracy.
  • Ration-legal authority depends on routinized administration, which often involves a lot of paperwork.
  • According to Weber, rational-legal authority is a form of leadership in which the authority of an organization or a ruling regime is largely tied to legal rationality, legal legitimacy, and bureaucracy.

• Rational Inequalities

  • As with solving polynomial inequalities, the first step to solving rational inequalities is to find the zeros.  
  • Because a rational expression consists of the ratio of two polynomials, the zeroes for both polynomials will be needed.
  • Graph of a rational polynomial with the equation $y=\frac{x^2+2x-3}{x^2-4}$.
  • For $x$ values that are zeros for the numerator polynomial, the rational function overall is equal to zero.
  • Solve for the zeros of a rational inequality to find its solution

• Rational discourse

  • "Rational discourse is a catalyst for transformation, as it induced the various participants to explore the depth and meaning of their various world-views, and articulate those ideas to their instructor and classmates" (Mezirow, 1991).
  • This debate was a good example of rational discourse and an essential factor for changing meaning perspective because they are engaged in sharing their own feelings on their current view of AutoCAD.

• Integer Coefficients and the Rational Zeroes Theorem

  • In algebra, the Rational Zero Theorem, or Rational Root Theorem, or Rational Root Test, states a constraint on rational solutions (also known as zeros, or roots) of the polynomial equation
  • Since any integer has only a finite number of divisors, the rational root theorem provides us with a finite number of candidates for rational roots.
  • Since every polynomial with rational coefficients can be multiplied with an integer to become a polynomial with integer coefficients and the same zeroes, the Rational Root Test can also be applied for polynomials with rational coefficients.
  • We can use the Rational Root Test to see whether this root is rational.
  • Use the Rational Zeros Theorem to find all possible rational roots of a polynomial

• Rational Equations

  • A rational equation is when two rational expressions are set equal to each other and unknown values are found that make the equation true.
  • In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable.
  • Notice that the rational expressions on both sides of the equal sign have the same denominator?
  • If you have a rational equation where the denominators on either side of the equation are the same, then their respective numerators must be the same value, even though they might be expressed differently.  
  • Solve the rational equation by cross-multiplication; therefore, $10\cdot x=14\cdot 17$, or $10x=238$.