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Chapter 2: Problem 90

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$h(x)=2 x^{4}-3 x+2$$

Short Answer

Expert verified
The polynomial \(h(x) = 2x^4 - 3x + 2\) based on Descartes's Rule of Signs potentially has one or zero positive real zeros and zero negative real zeros.

Step by step solution

01

Determine the possible number of positive real zeros

Count the number of sign changes in the polynomial \(h(x) = 2x^4 - 3x + 2\). There is only one sign change (from \(+2x^4\) to \(-3x\)), so there can be either one or zero positive real zeros.
02

Determine the possible number of negative real zeros

Substitute \(x\) with \(-x\) in the polynomial \(h(x)\) and count the number of sign changes. The result is \(h(-x) = 2(-x)^4 - 3(-x) + 2 = 2x^4 + 3x + 2\). There are no sign changes in \(h(-x)\), so there are no negative real zeros.
03

Summary of Findings

In summary, using Descartes's Rule of Signs, we find that \(h(x) = 2x^4 - 3x + 2\) possibly has one positive real zeros and zero negative real zeros.

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Most popular questions from this chapter

Think About It Let \(y=f(x)\) be a cubic polynomial with leading coefficient \(a=-1\) and \(f(2)=f(i)=0\) Write an equation for \(f\)

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-2 x$$

Sketch the graph of each polynomial function. Then count the number of real zeros of the function and the numbers of relative minima and relative maxima. Compare these numbers with the degree of the polynomial. What do you observe? (a) \(f(x)=-x^{3}+9 x\) (b) \(f(x)=x^{4}-10 x^{2}+9\) (c) \(f(x)=x^{5}-16 x\)

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

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