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Chapter 2: Problem 27

Find the rational zeros of the function. $$f(x)=9 x^{4}-9 x^{3}-58 x^{2}+4 x+24$$

Short Answer

Expert verified
The rational zeros of the function \(f(x) = 9x^4 - 9x^3 - 58x^2 + 4x + 24\) are \(x = 1\) and \(x = -2/3\).

Step by step solution

01

List out Factors

We first make a list of the factors for the constant term and the leading coefficient. The factors of the constant term \(24\) are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\). The factors of the leading coefficient \(9\) are \(\pm 1, \pm 3, \pm 9\).
02

Formulate Potential Rational Zeros

Then we form the potential rational zeros using every combination of a factor of the constant term divided by a factor of the leading coefficient. The possible rational zeros are hence: \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24, \pm 1/3, \pm 2/3, \pm 4/3, \pm 8/3, \pm 12/3, \pm 24/3, \pm 1/9, \pm 2/9, \pm 1/3, \pm 4/9, \pm 8/9, \pm 2/3, \pm 12/9, \pm 24/9.\)
03

Plugging in Potential Zeroes

Now, we evaluate the function \(f(x)\) at each of these potential zeros. The function \(f(x)\) equals zero at \(x = 1, x = -2/3\). Therefore, these are the rational zeros of the function \(f(x) = 9x^4 - 9x^3 - 58x^2 + 4x + 24\)

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Most popular questions from this chapter

Determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\) $$g(x)=f(2 x)$$

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