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Chapter 2: Problem 93

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

Short Answer

Expert verified
The function could have either 2 or 0 positive real zeros, and exactly 1 negative real zero.

Step by step solution

01

Identifying sign changes in the original function

Observe the initial function \(f(x)=-5x^3 + x^2 - x + 5\). The changes in sign occur from the first term to the second term \(-5x^3\) to \(x^2\), and from the third term to the fourth term \(-x\) to \(5\). Thus, there are 2 sign changes in the original function.
02

Identifying sign changes in the function with replaced 'x' with '-x'

Replace \(x\) with \(-x\) in the equation, which leads to: \(f(-x)=-5(-x)^3 + (-x)^2 - (-x) + 5 = -5x^3 + x^2 + x + 5\). There is just 1 sign change in this modified function, occurring from the second term to the third term \(x^2\) to \(x\).
03

Applying Descartes's Rule of Signs

According to Descartes's Rule of Signs, the number of positive real zeros is equal to the number of sign changes in the original function or less by an even number. Similarly, the number of negative real zeros is equal to the number of sign changes in the function where \(x\) has been replaced by \(-x\) or less by an even number.

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

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

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Use the given zero to find all the zeros of the function. Function \(f(x)=2 x^{3}+3 x^{2}+18 x+27\) Zero \(3 i\)

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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