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Chapter 2: Problem 90

Use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero. $$f(x)=\frac{1}{4} x^{4}-2 x^{2}$$

Short Answer

Expert verified
The zeros of the function \(f(x) = \frac{1}{4} x^{4} - 2 x^{2}\) are x=-2, x=0, and x=2. The multiplicities of these zeros respectively are 1, 2, and 1.

Step by step solution

01

Graph the function

To graph this function, it can be plugged into a graphing utility like a graphing calculator or an online graphing tool. The graph will visually represent the function.
02

Approximate the real zeros

By looking at the graph, the real zeros of the function can be approximately determined. These are the x-values where the function crosses or touches the x-axis. For this function, the real zeros appear to occur at x=-2, x=0, and x=2.
03

Determine the multiplicity of each zero

The multiplicity of a zero is determined by the power of the factor in the factored form of the polynomial. The zero x=0 occurs twice in the polynomial (as \(x^2\) and \(x^4\)), therefore it has a multiplicity of 2. The zeros x=-2 and x=2 occur once, so they both have a multiplicity of 1.

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Most popular questions from this chapter

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