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Chapter 2: Problem 8

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$f(x)=4 x^{5}-8 x^{4}-x+2$$

Short Answer

Expert verified
The possible rational zeros of the function \(f(x)= 4x^{5} - 8x^{4} - x + 2\) are \(\pm1, \pm1/2, \pm1/4, \pm2\).

Step by step solution

01

Identify the Coefficients

The leading coefficient is the coefficient of the highest degree term, which in this case is \(4\) for the term \(4x^{5}\). The constant term is the term with no \(x\), which here is \(2\).
02

Find the Factors

Find all the factors of \(4\), the leading coefficient, and \(2\), the constant term. The factors of \(4\) are \(\pm1, \pm2, \pm4\). The factors of \(2\) are \(\pm1, \pm2\).
03

List all Possible Rational Zeros

Using the Rational Zero Theorem, form the fractions \(\pm p/q\) where \(p\) are factors of the constant term and \(q\) are factors of the leading coefficient. The possible rational zeros will therefore be \(\pm1/1, \pm1/2, \pm1/4, \pm2/1, \pm2/2, \pm2/4\). This simplifies to \(\pm1, \pm1/2, \pm1/4, \pm2, \pm1, \pm1/2\). Removing duplicates gives the final list: \(\pm1, \pm1/2, \pm1/4, \pm2\).

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