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Chapter 2: Problem 26

Find the rational zeros of the function. $$f(x)=3 x^{3}-19 x^{2}+33 x-9$$

Short Answer

Expert verified
The rational zeros of the function \(f(x)=3x^3-19x^2+33x-9\) are -1, 1/3, and 3.

Step by step solution

01

Identify the possible rational roots

Using the Rational Root Theorem, identify the possible rational zeros. This involves identifying the factors of the constant term -9 (which are \( \pm 1, \pm 3, \pm 9 \)) and the leading coefficient 3 (which are \( \pm 1, \pm 3 \)). Generate the list of possible zeros by dividing every factor of the constant term by every factor of the leading coefficient. This gives the list \( \pm 1, \pm 3, \pm 9, \pm 1/3, \pm 3/3, \pm 9/3 \). Removing the repetitions, we get the list of possible rational zeros: \( \pm 1, \pm 3, \pm 9, \pm 1/3 \)
02

Use synthetic division or the Remainder Theorem

To verify which of these possible zeros are in fact zeros of the function, use synthetic division or the Remainder Theorem. This means we substitute each of the possible zeros into the polynomial function and see if the result equals zero.
03

Verify the possible zeros

By substituting the possible zeros \( \pm 1, \pm 3, \pm 9, \pm 1/3 \) into the function \(f(x)=3x^{3}-19x^{2}+33x-9 \), we compute and compare the results with zero. After checking all possibilities, we find that \( f(-1)=0 \), \( f(1/3)=0 \) and \(f(3)=0\), therefore the rational zeros of the function are -1, 1/3, and 3.

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