91Ó°ÊÓ

A rational function is a fraction where both the numerator and the denominator are polynomials. The general form is \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials.
Remember, the domain of a rational function includes all real numbers except those that make the denominator zero. Finding the domain starts with identifying the denominator of the function. If the denominator equals zero for a certain \(x\) value, then the function is not defined at that point and must be excluded from the domain.
For example, in \(f(x) = \frac{x^4 + 7}{x^2 + 6x + 5}\), the denominator is \(x^2 + 6x + 5\). We need to find out where this denominator becomes zero by solving \(x^2 + 6x + 5 = 0\).

Quadratic equations are polynomials of degree two and have the general form \(ax^2 + bx + c = 0\). Solving quadratic equations often involves factoring, using the quadratic formula, or by completing the square.
In our example, the denominator of the rational function is \(x^2 + 6x + 5\), a quadratic equation. To find the values of \(x\) that make this equation zero, you can factorize it. If \(x^2 + 6x + 5\) is factored into \( (x + 1)(x + 5)\), we set each factor to zero, yielding solutions for \(x\): \(x = -1\) and \(x = -5\).

Factoring is breaking down a complex expression into simpler ones that, when multiplied together, give the original expression. For quadratic equations, this often means finding two binomials that multiply to give the quadratic.
To factorize \(x^2 + 6x + 5\), find two numbers that multiply to 5 (the constant term) and add up to 6 (the coefficient of \(x\)). Here, those numbers are 1 and 5, leading to the factors \( (x + 1)(x + 5)\).
Setting these factors equal to zero gives us the values \(x = -1\) and \(x = -5\), helping us determine where the function is not defined, thus indicating the domain of the rational function: all real numbers except \(x = -1\) and \(x = -5\).