What makes [tex]$\frac{p-1}{p(p-2)}$[/tex] undefined?
Answer (2)
The expression is undefined when the denominator equals zero.
Set the denominator p ( p − 2 ) to zero.
Solve for p , which gives p = 0 or p = 2 .
The expression is undefined when p = 0 or p = 2 .
Explanation
Understanding the Problem We are asked to find the values of p that make the expression p ( p − 2 ) p − 1 undefined. A rational expression is undefined when the denominator is equal to zero.
Setting the Denominator to Zero To find the values of p that make the expression undefined, we need to set the denominator equal to zero and solve for p . The denominator is p ( p − 2 ) . So we have the equation: p ( p − 2 ) = 0
Solving for p We can solve this equation by setting each factor equal to zero:
p = 0 or p − 2 = 0
If p − 2 = 0 , then p = 2 .
Final Answer Therefore, the expression p ( p − 2 ) p − 1 is undefined when p = 0 or p = 2 .
Examples
Understanding when a rational expression is undefined is crucial in various fields, such as physics and engineering, where mathematical models often involve rational functions. For instance, when analyzing electrical circuits, the impedance of a circuit can be represented as a rational function of frequency. Identifying the values of frequency that make the impedance undefined helps engineers avoid resonance conditions that could damage the circuit. Similarly, in fluid dynamics, the velocity field of a fluid can be modeled using rational functions, and identifying points where the denominator becomes zero helps predict singularities or points of instability in the flow.
The expression p ( p − 2 ) p − 1 is undefined when the denominator equals zero. This occurs at p = 0 and p = 2 . Therefore, the expression is undefined at these values.
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