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Chapter 2: Problem 91

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$g(x)=5 x^{5}-10 x$$

Short Answer

Expert verified
The function \(g(x) = 5 x^{5}-10x\) has one positive real zero and no negative real zeros as per Descartes' Rule of Signs.

Step by step solution

01

Identify the function and its coefficients

The function given is \(g(x) = 5 x^{5}-10x\). Its coefficients are 5 for \(x^{5}\), 0 for \(x^{4}\), 0 for \(x^{3}\), 0 for \(x^{2}\), -10 for \(x\) and 0 for the constant term.
02

Apply Descartes' Rule for positive real zeros

Apply Descartes' Rule, look for sign changes in the list of coefficients. Here there is only one sign change (from 5 to -10). Therefore, there is exactly one positive real zero.
03

Apply Descartes' Rule for negative real zeros

Next, apply Descartes' Rule to find the possible negative real zeros. This is done by replacing x with -x in the polynomial and count the number of sign changes again. The function becomes \(g(-x) = -5x^{5} -10x\). In this case, there are no changes in sign, so there are no negative real zeros.

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Most popular questions from this chapter

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

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 $$f(x)=x^{3}-2 x$$

Find all real zeros of the function. $$f(z)=12 z^{3}-4 z^{2}-27 z+9$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{2}+36$$

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-3 x^{3}-x^{2}-12 x-20\) (Hint: One factor is \(\left.x^{2}+4 .\right)\)
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