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Rational functions are expressions formed as a ratio of two polynomials. In other words, they are fractions where the numerator and the denominator are both polynomial functions. For example, in the function given in the exercise, \(f(x) = \frac{x^2 - 2x}{x^2 - 3x + 2}\), the numerator is \(x^2 - 2x\) and the denominator is \(x^2 - 3x + 2\). Rational functions can be quite complex because they involve division, which opens up the potential for discontinuities.

A critical aspect to understand about rational functions is that they are undefined wherever the denominator is equal to zero. This is because dividing by zero is mathematically undefined. As such, finding where the denominator equals zero is crucial for identifying points where the function might not exist or might exhibit infinite behavior. It's in these places we often find discontinuities, after ensuring there are no simplifying cancellations with the numerator.

• Rational functions can be simplified by factoring both the numerator and denominator and canceling common factors, though simplification isn't always possible.
• Discontinuities are often indicated by zeroes of the denominator. These discontinuities can be either holes or vertical asymptotes.
• When the numerator and denominator share a factor, it could lead to a removable discontinuity, sometimes referred to as a hole in the graph.

Factoring quadratic expressions is a fundamental technique used in dealing with polynomials, especially when identifying discontinuities in rational functions. A quadratic expression is generally of the form \(ax^2 + bx + c\) and can often be factored into two binomials, providing it simplifies nicely.

The process of factoring a quadratic involves finding two numbers that multiply to give the constant term \(c\), and add up to give the coefficient of the middle term \(b\). For example, in the denominator \(x^2 - 3x + 2\), we look for numbers that multiply to 2 and add up to -3. These numbers are -1 and -2, allowing us to factor the expression as \((x - 1)(x - 2)\).

• Factoring reveals the zeroes of the polynomial by setting each factor equal to zero and solving for \(x\).
• This method also simplifies expressions and can assist in finding points of cancellation in rational functions.
• Mastering factoring is essential for effectively managing polynomial equations, inequalities, and expressions.

Finding the zeroes of polynomials is a crucial step in analyzing polynomial functions, particularly when working with rational functions. The zeroes of a polynomial are the values of \(x\) that make the polynomial equal to zero. These are also called the roots or solutions of the polynomial.

When dealing with a factored polynomial, finding its zeroes becomes straightforward. Simply take each factor of the polynomial, set it equal to zero, and solve for \(x\). In the expression \((x - 1)(x - 2) = 0\), set each factor equal to zero: \(x - 1 = 0\) or \(x - 2 = 0\). Solving these yields the zeroes \(x = 1\) and \(x = 2\).

• Zeroes indicate the x-intercepts of the graph if the polynomial is set as a standalone equation.
• In rational functions, the zeroes of the denominator can indicate points of discontinuity.
• These zeroes need to be examined contextually to understand their impact, such as checking if they cause a hole or a vertical asymptote in the graph.