Problem 27 - Elementary Algebra

Question

For the following problems, find the domain of each of the rational expressions. $$ \frac{y-9}{y^{2}-y-20} $$

Step-by-Step Solution

Answer

Answer: The domain of the rational expression is $$(-\infty,-4) \cup (-4, 5) \cup (5, \infty)$$, excluding the values y = 5 and y = -4.

Step 1: Identify the denominator

In this expression, the denominator is: $$ y^2 - y - 20 $$

Step 2: Find the values of y that make the denominator zero

To do this, we need to solve the following equation for y: $$ y^2 - y - 20 = 0 $$ To solve this quadratic equation, we can factor it into two binomials: $$ (y - 5)(y + 4) = 0 $$ Now we can set both factors to zero and solve for y: $$ y - 5 = 0 \Rightarrow y = 5 \\ y + 4 = 0 \Rightarrow y = -4 $$

Step 3: Write down the domain

The domain of the rational expression is all real numbers excluding the values of y that make the denominator zero. In this case, y = 5 and y = -4. Thus, the domain of the expression is: $$ (-\infty,-4) \cup (-4, 5) \cup (5, \infty) $$

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential skill in algebra, particularly when it comes to simplifying expressions or solving equations. A quadratic is a polynomial of the form \( ax^2 + bx + c \). For many quadratics, especially when \( a = 1 \), the equation can be factored into the form \((x - r)(x - s)\), where \( r \) and \( s \) are the solutions to the equation.To factor a quadratic like \( y^2 - y - 20 \), you need to find two numbers that multiply to \(-20\) (the constant term) and add up to \(-1\) (the coefficient of the linear term). Here are the steps to factor:

• Identify the constant term (\(-20\)) and the linear coefficient (\(-1\)).
• Think of two numbers that multiply to \(-20\) and add to \(-1\). Those numbers will be \(-5\) and \(4\).
• Rewrite the quadratic as \((y - 5)(y + 4)\).

Factoring is a helpful technique as it reveals the roots of the quadratic equation, allowing us to solve for the variable and find critical values, such as those that might make a denominator zero in a rational expression.

Rational Functions

Rational functions are functions that can be expressed as the quotient of two polynomials. Generally, they are written in the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). It’s important to remember that the domain of a rational function excludes values that would make the denominator zero.Understanding rational functions involves:

• Identifying the numerator and the denominator functions.
• Determining where the denominator \( Q(x) \) becomes zero, as these points are undefined in the domain.

For instance, in the rational expression \( \frac{y-9}{y^{2}-y-20} \), the critical task is to check where the denominator \( y^2 - y - 20 \) equals zero. Factoring, as discussed in the previous section, aids in this process by simplifying the identification of excluded values.

Excluded Values in Domain

Excluded values in the domain of a rational function arise from points where the function is undefined. This typically happens when the denominator is zero, as division by zero is undefined in mathematics. To find these excluded values:

• Solve the equation \( Q(x) = 0 \), where \( Q(x) \) is the denominator of the rational function.
• These solutions are the values of the variable that need to be excluded from the domain.

For the expression \( \frac{y-9}{y^{2}-y-20} \), we solved \( y^2 - y - 20 = 0 \) to find \( y = 5 \) and \( y = -4 \). Thus, the domain excludes these values:\[ (-\infty,-4) \cup (-4, 5) \cup (5, \infty) \]Remember, excluding these values ensures we preserve the function's validity since where the denominator is zero, the function doesn't exist.