An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Factor the polynomial: f ( x ) = − 2 ( x − 3 ) 2 ( x − 5 ) ( x + 5 ) .
Identify the multiplicity of each zero: x = − 5 (multiplicity 1), x = 3 (multiplicity 2), x = 5 (multiplicity 1).
Determine if the graph crosses or touches the x-axis at each zero based on its multiplicity.
State the behavior of the graph at each x-intercept: crosses at x = − 5 , touches at x = 3 , crosses at x = 5 .
Explanation
Understanding the Problem We are given the polynomial function f ( x ) = − 2 ( x − 3 ) 2 ( x 2 − 25 ) . We need to determine the multiplicity of each real zero and whether the graph crosses or touches the x-axis at each x-intercept.
Factoring the Polynomial First, we factor the polynomial completely. We know that x 2 − 25 can be factored as ( x − 5 ) ( x + 5 ) . So, we have f ( x ) = − 2 ( x − 3 ) 2 ( x − 5 ) ( x + 5 ) .
Identifying Zeros and Multiplicities Now we identify the zeros and their multiplicities:
The zero x = − 5 comes from the factor ( x + 5 ) , which has a power of 1. So, the multiplicity of x = − 5 is 1.
The zero x = 3 comes from the factor ( x − 3 ) 2 , which has a power of 2. So, the multiplicity of x = 3 is 2.
The zero x = 5 comes from the factor ( x − 5 ) , which has a power of 1. So, the multiplicity of x = 5 is 1.
Determining Graph Behavior at Zeros Next, we determine whether the graph crosses or touches the x-axis at each zero. If the multiplicity is odd, the graph crosses the x-axis. If the multiplicity is even, the graph touches the x-axis.
At x = − 5 , the multiplicity is 1, which is odd. So, the graph crosses the x-axis at x = − 5 .
At x = 3 , the multiplicity is 2, which is even. So, the graph touches the x-axis at x = 3 .
At x = 5 , the multiplicity is 1, which is odd. So, the graph crosses the x-axis at x = 5 .
Final Answer Therefore, the smallest real zero, -5, is a zero of multiplicity 1, so the graph of f crosses the x-axis at x = -5. The middle real zero, 3, is a zero of multiplicity 2, so the graph of f touches the x-axis at x = 3. The largest real zero, 5, is a zero of multiplicity 1, so the graph of f crosses the x-axis at x = 5.
Examples
Understanding the behavior of polynomial functions, such as where they cross or touch the x-axis, is crucial in many real-world applications. For example, in engineering, when designing a bridge, engineers need to analyze the load distribution, which can be modeled by polynomial functions. Knowing where the function equals zero (crosses the x-axis) helps identify critical points where the structure experiences maximum stress or deflection. Similarly, in economics, polynomial functions can model cost and revenue curves, and identifying the zeros helps determine break-even points where the company neither makes nor loses money. This analysis ensures safe and efficient designs and informed business decisions.
