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Chapter 2: Problem 91

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. $$g(x)=\frac{1}{5}(x+1)^{2}(x-3)(2 x-9)$$

Short Answer

Expert verified
The real zeros of the function \(g(x)=\frac{1}{5}(x+1)^{2}(x-3)(2x-9)\) are x = -1, x = 3, and x = 4.5. The multiplicity of x = -1 is 2, while the multiplicity of both x = 3 and x = 4.5 is 1 each.

Step by step solution

01

Graph the given function

Using a graphing utility, plot the function \(g(x)=\frac{1}{5}(x+1)^{2}(x-3)(2x-9)\). Look for the points where the graph crosses the x-axis. These points are the real zeros of the function.
02

Identify the real zeros

The graph crosses the x-axis at three points. These points correspond to the values of x that make the function equal to zero, and thus, they are the real zeros of the function.
03

Determine the multiplicity of each zero

Multiplicity refers to how many times the same root is repeated in a function. Notice that in the function, \((x+1)^{2}\) is a repeated root, while \(x-3\) and \(2x-9\) are unique roots. Based on this, x = -1 has multiplicity of 2, while x = 3 and x = 4.5 each have a multiplicity of 1. This indicates that the graph touches and bounces off the x-axis at x = -1, and cuts through the x-axis at x = 3 and x = 4.5.

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Most popular questions from this chapter

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=2 x^{4}+5 x^{3}+4 x^{2}+5 x+2$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(y)=y^{4}-256$$

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=-5 x^{3}+x^{2}-x+5$$

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=2 x^{4}-8 x+3\) (a) Upper: \(x=3\) (b) Lower: \(x=-4\)

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