Problem 6 - Algebra and Trigonometry

Question

find all numbers that must be excluded from the domain of each rational expression. $$ \frac{x-3}{x^{2}+4 x-45} $$

Step-by-Step Solution

Answer

The numbers that must be excluded from the domain of the given rational expression are \(x = 5\) and \(x = -9\).

Step 1: Identify the Denominator

The denominator of the rational expression in the exercise is \(x^2 + 4x - 45\). This is a quadratic equation and the requirement for any rational expression is that this cannot be equal to zero.

Step 2: Find the Roots of the Denominator

Set the denominator equal to zero and solve for x: \(x^2 + 4x - 45 = 0\). One way to solve this is by factoring. The two numbers that add up to 4 (the coefficient of x) and multiply to -45 are 9 and -5. Thus, the factored form of the quadratic is \((x - 5)(x + 9) = 0\). This equation is true if either \(x - 5 = 0\) or \(x + 9 = 0\). Solving for x yields \(x = 5\) or \(x = -9\).

Step 3: Determine the Excluded Values

The values that we found to make the denominator zero (i.e. \(x = 5\) and \(x = -9\)) are the ones that we exclude from the domain, because they would make the given rational expression undefined.

Key Concepts

Solving Quadratic Equations

Quadratic equations are a staple in algebra and represent a significant area of study in mathematics. These equations are in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). In our exercise, the quadratic equation is the denominator \(x^2 + 4x - 45\).

To solve these equations, we have several methods at our disposal, such as factoring, completing the square, or using the quadratic formula. The simplest and often preferred method is factoring, which is applicable if the quadratic is factorable. For instance, \(x^2 + 4x - 45\) can be expressed as \(x - 5)(x + 9)\), indicating that the solutions are \(x = 5\) and \(x = -9\).
Many students find this method more intuitive, as it connects to the fundamental property that if the product of two factors is zero, at least one of the factors must be zero. This principle leads to finding the solutions by setting each factor equal to zero. It's worth noting that not all quadratics are factorable, which would necessitate using other solving techniques.

Factoring Polynomials

Factoring polynomials is a critical skill that allows us to simplify expressions and solve equations efficiently. When faced with a polynomial like the quadratic in our exercise, the goal is to rewrite it as a product of simpler polynomials, its factors, which are easier to manage.

In the given example, \(x^2 + 4x - 45\), we look for two numbers that multiply to give the constant term (\(-45\)) and add up to give the coefficient of the \(x\) term (4). The numbers 9 and -5 work nicely since \(9 \times (-5) = -45\) and \(9 + (-5) = 4\).
The polynomial then factors into \( (x - 5)(x + 9) \). This method hinges on recognizing patterns and pairs of numbers that work together, an important tool not only for quadratic equations but for polynomials of higher degrees as well. By breaking down more complex polynomials into easier pieces, students can solve a variety of mathematical problems more effectively.

Excluded Values in Rational Expressions

In algebra, rational expressions are fractions that contain polynomials in their numerator and denominator. An essential idea in handling these expressions is to identify their domain, which represents all the possible values that can be substituted for the variable.

Excluded values are numbers that would make the denominator of a rational expression equal to zero, which is not allowed, as it would make the expression undefined. For our exercise, the denominator monomial \(x^2 + 4x - 45\) dictates the domain. We exclude the values \(x = 5\) and \(x = -9\) because substituting them into the denominator would result in zero.
Understanding excluded values is crucial for proper graphing, simplifying, and working with rational expressions in calculus and higher-level math courses. Students should remember to check for these values to avoid mistakes, especially when solving equations or inequalities involving rational expressions.