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Factoring quadratics is a crucial skill when dealing with polynomial functions. A quadratic expression often appears in the form of \( ax^2 + bx + c \). To factor it, we look for two binomials \((px + q)(rx + s)\) that multiply to give the original quadratic. The process requires you to find two numbers that not only add to give the middle term (the coefficient of \(x\)) but also multiply to give the constant term (the product \(c\)).
In the case of \(x^2 - 2x - 15\), we need to find two numbers that multiply to \(-15\) and add to \(-2\). These numbers are \(-5\) and \(+3\). Therefore, \(x^2 - 2x - 15\) can be factored as \((x - 5)(x + 3)\).
Understanding how to factor quadratics will help you simplify functions and solve equations effectively. With practice, recognizing patterns and applying the appropriate method will become second nature.

A function becomes undefined whenever its operations involve division by zero. This can happen in rational functions that have a polynomial in the denominator. To find out where a function is undefined, set the denominator equal to zero and solve for \(x\). These \(x\) values are where the function does not exist as they would cause division by zero.
For example, given the function \(g(x) = \frac{3}{x^2 - 2x - 15}\), we've factorized the denominator to \((x - 5)(x + 3)\). Setting each factor equal to zero gives \(x = 5\) and \(x = -3\). This means \(g(x)\) is undefined for these two \(x\) values because substituting them into the denominator results in zero. It’s essential to identify these undefined points to correctly determine the domain of a function.

Real numbers encompass a wide range of numerical values that we use in everyday mathematics. They include all rational numbers, such as integers and fractions, and irrational numbers like \(\sqrt{2}\) or \(\pi\). Essentially, any number that can be placed on a number line is considered a real number.
When determining the domain of a function, we typically consider all real numbers. However, we exclude numbers that cause the function to be undefined. In functions like \(g(x) = \frac{3}{x^2 - 2x - 15}\), all real numbers can be part of the domain except for those that make the denominator zero (as these points make the function undefined). Thus, for \(g(x)\), the domain includes all real numbers except for \(x = 5\) and \(x = -3\).
Understanding the scope and limitations of real numbers helps in comprehending more complex mathematical concepts and ensuring accurate computations with functions.