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Chapter 2: Problem 89

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. $$f(x)=x^{3}-16 x$$

Short Answer

Expert verified
The function \(f(x) = x^{3} - 16x\) has zeros at \(x = -4, 0, 4\). Each of these zeros has a multiplicity of 1, since the graph crosses the x-axis at these points.

Step by step solution

01

Graphing the function

Use a graphing utility (like Desmos, GeoGebra or a graphing calculator) to plot the function \(f(x) = x^{3} - 16x\). Observe where the curve of the function intersects with the x-axis, as these points represent the possible zeros of the function.
02

Find the zeros

Use the zero or root feature of your graphing utility to approximate the real zeros of the function. These are the x-values where the graph intersects the x-axis.
03

Determine the Multiplicity

The multiplicity of a zero refers to how many times it occurs as a root in the function. This can often be inferred from the graph. If the graph crosses the x-axis at a zero, then that zero has an odd multiplicity. If the graph touches the x-axis and then turns in the opposite direction at a zero, then that zero has an even multiplicity. Calculate this accordingly for each zero.

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Most popular questions from this chapter

Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) $$5,3-2 i$$

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{1}{4} x^{2}-x+\frac{1}{4}=\frac{1}{4}\left(4 x^{3}-x^{2}-4 x+1\right)$$

Determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\) $$g(x)=3 f(x)$$

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Use the given zero to find all the zeros of the function. Function \(g(x)=x^{3}-7 x^{2}-x+87\) Zero \(5+2 i\)
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