To find all possible rational zeros of the function \( f(x) = 6x^4 - 3x^2 + 2 \) using the Rational Zero Theorem, we need to examine the factors of the constant term and the leading coefficient.
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Identify the constant term and leading coefficient:
- The constant term (the term without \( x \)) is \( 2 \).
- The leading coefficient (the coefficient of the highest degree term \( x^4 \)) is \( 6 \).
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List the factors of the constant term \( 2 \):
- The factors of \( 2 \) are \( \pm 1, \pm 2 \).
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List the factors of the leading coefficient \( 6 \):
- The factors of \( 6 \) are \( \pm 1, \pm 2, \pm 3, \pm 6 \).
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Form possible rational zeros by taking all combinations of the factors of the constant term over the factors of the leading coefficient:
- Possible rational zeros are:
- \( \frac{\text{factor of constant}}{\text{factor of leading coefficient}} \)
This leads to the following combinations:
- \( \pm 1 = \frac{1}{1} \) and \( \frac{-1}{1} \)
- \( \pm 2 = \frac{2}{1} \) and \( \frac{-2}{1} \)
- \( \pm \frac{1}{2} = \frac{1}{2} \) and \( \frac{-1}{2} \)
- \( \pm \frac{1}{3} = \frac{1}{3} \) and \( \frac{-1}{3} \)
- \( \pm \frac{1}{6} = \frac{1}{6} \) and \( \frac{-1}{6} \)
- \( \pm \frac{2}{3} = \frac{2}{3} \) and \( \frac{-2}{3} \)
- \( \pm 3 = \frac{3}{1} \) and \( \frac{-3}{1} \)
- \( \pm 6 = \frac{6}{1} \) and \( \frac{-6}{1} \)
- Possible rational zeros are:
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Combining all these, we have the list of possible rational zeros:
- \( \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{2}{3}, \pm 3, \pm 6 \)
Thus, the correct response from your options can be narrowed down to:
- Response: \( \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{2}{3} \)