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Chapter 3: Problem 77

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros of \(f(x)-x^{5}-x^{4}+x^{3}-x^{2}+x-8 .\) Verify your result by using a graphing utility to graph \(f\)

Short Answer

Expert verified
The possible number of positive real zeros is 1, 3, or 5 and the possible number of negative real zeros is 0 or 2. This can be verified using a graphing utility.

Step by step solution

01

Determine possible number of positive real zeros

Replace \(x\) in \(f(x)=x^{5}-x^{4}+x^{3}-x^{2}+x-8\) by \(+x\) and count the number of sign changes to get the possible number of positive real zeros. The sign sequence is (+,-,+,-,+,-), therefore there are 5 sign changes. So, there are 5 or 5-2=3 or 3-2=1 positive real zeros.
02

Determine possible number of negative real zeros

Now replace \(x\) in \(f(x)=x^{5}-x^{4}+x^{3}-x^{2}+x-8\) by \(-x\), to get the possible number of negative real zeros. This gives \(-x^5 - x^4 - x^3 + x^2 - x + 8\). The sign sequence is (-,-,-,+,-,+), resulting in 2 sign changes. So, there could be 2 or 2-2=0 negative real roots.
03

Verify with a Graphing Utility

You can confirm your results using a graphing utility. Plot the function \(f(x)=x^{5}-x^{4}+x^{3}-x^{2}+x-8\), and observe the number of times the graph intersects the x-axis. This represents the number of real roots. Compare this with your calculated results for verification.

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Most popular questions from this chapter

Describe how to find a parabola's vertex if its equation is expressed in standard form. Give an example.

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose \(y\) -coordinate is the same as the given point. $$f(x)=(x-3)^{2}+2 ; \quad(6,11)$$

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y=-4 x^{2}+20 x+160$$

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