Problem 35 - Algebra 2

Question

Simplify each expression. For what values is \(\frac{2 d(d+1)}{(d+1)\left(d^{2}-4\right)}\) undefined?

Step-by-Step Solution

Answer

Undefined for \(d = -1, 2, -2\).

Step 1: Identify the Denominator

To determine when the expression is undefined, we need to focus on the denominator, which is \( (d+1)(d^2-4) \).

Step 2: Set Denominator Equal to Zero

Expressions are undefined when the denominator equals zero. So we set \( (d+1)(d^2-4) = 0 \).

Step 3: Solve for d+1 = 0

For the factor \( d+1 \), the expression is zero when \( d+1 = 0 \). Solving this, we get \( d = -1 \).

Step 4: Solve for d^2-4 = 0

The expression \( d^2-4 \) is zero when \( d^2 = 4 \). Taking the square root of both sides, we find \( d = 2 \) and \( d = -2 \).

Step 5: Compile All Values

The expression is undefined for \( d = -1, 2, -2 \) because these values make the denominator equal zero.

Key Concepts

Identifying Denominator

When simplifying rational expressions, identifying the denominator is a crucial first step. The denominator is the bottom part of a fraction and determines the values which can make the expression undefined. In the given exercise, the denominator is represented by \[ (d+1)(d^2-4) \]. It includes two factors, \(d+1\) and \(d^2-4\). Each factor can independently affect where the expression becomes undefined. Recognizing the complete denominator helps in pinpointing these values accurately.

Setting Denominator Equal to Zero

To find where a rational expression becomes undefined, we set the denominator equal to zero. An expression is undefined when the denominator is zero because division by zero is undefined in mathematics. Here, we set the equation \[ (d+1)(d^2-4) = 0 \]. Since the product of the factors is zero, at least one factor must itself be zero. This means we can solve each factor independently to determine the values that make the entire denominator zero. Solve \( d+1 = 0 \) and\( d^2-4 = 0 \).

Solving Quadratic Equations

In the context of this exercise, solving the equation \(d^2-4=0\) is required. This is a quadratic equation, which is an equation of the form \(ax^2+bx+c=0\). For such equations, you can use factoring, completing the square, or the quadratic formula to find solutions. To solve \(d^2-4=0\), notice that it can be factored as a difference of squares: \((d-2)(d+2)=0\). Setting each factor to zero, we solve \(d-2=0\) giving \(d=2\) and \(d+2=0\) resulting in \(d=-2\). Therefore, the solutions are \(d=2\) and \(d=-2\). These are critical in determining undefined values.

Expression Undefined Values

After calculating the values that make the denominator zero, we identify the values for which the expression is undefined. In our exercise, the undefined values were found by solving \(d+1=0\) and \(d^2-4=0\), resulting in the values \(d=-1, 2,\) and \(-2\). Each of these values makes one or more factors of the denominator equal zero, rendering the expression undefined. Listing these values helps avoid errors in calculation and ensures that the rational expression is only used for permissible inputs. Remember, an expression is undefined exactly where its denominator is zero, which is why finding these values is essential.