To apply the Rational Zero Theorem, we need to identify the possible rational roots of the polynomial function \( f(x) = 14x^7 - 4x^2 + 2 \).
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Identify the leading coefficient and the constant term:
- Leading coefficient (the coefficient of the highest degree term): \( 14 \)
- Constant term: \( 2 \)
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Determine the factors of the leading coefficient (14):
- Factors of \( 14 \): \( \pm 1, \pm 2, \pm 7, \pm 14 \)
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Determine the factors of the constant term (2):
- Factors of \( 2 \): \( \pm 1, \pm 2 \)
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Form the possible rational zeros: The possible rational zeros are given by the ratios of the factors of the constant term over the factors of the leading coefficient. This results in:
\[ \text{Possible rational zeros} = \frac{\text{factors of constant term}}{\text{factors of leading coefficient}} \]
Listing these:
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From \( \pm 1 \):
- \( \frac{1}{1} = \pm 1 \)
- \( \frac{1}{2} = \pm \frac{1}{2} \)
- \( \frac{1}{7} = \pm \frac{1}{7} \)
- \( \frac{1}{14} = \pm \frac{1}{14} \)
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From \( \pm 2 \):
- \( \frac{2}{1} = \pm 2 \)
- \( \frac{2}{2} = \pm 1 \) (already included)
- \( \frac{2}{7} = \pm \frac{2}{7} \)
- \( \frac{2}{14} = \pm \frac{1}{7} \) (already included)
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Thus, when you collect all distinct possibilities, we have: \[ \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{7}, \pm \frac{1}{14}, \pm \frac{2}{7} \]
Now let's compare this with the provided responses:
The correct response would be:
- plus or minus \( 1 \), plus or minus \( 2 \), plus or minus \( \frac{1}{2} \), plus or minus \( \frac{1}{7} \), plus or minus \( \frac{1}{14} \), plus or minus \( \frac{2}{7} \).
Hence, this corresponds to the first option you provided.