Question

Use the Rational Zero Theorem to create a list of all possible rational zeroes of the function f left parenthesis x right parenthesis equals 14 x superscript 7 baseline minus 4 x squared plus 2.
(1 point)
Responses

plus or minus 1 comma plus or minus 2 comma plus or minus one half comma plus or minus Start Fraction 1 over 7 End Fraction comma plus or minus Start Fraction 1 over 14 End Fraction comma plus or minus Start Fraction 2 over 7 End Fraction
Image with alt text: plus or minus 1 comma plus or minus 2 comma plus or minus one half comma plus or minus Start Fraction 1 over 7 End Fraction comma plus or minus Start Fraction 1 over 14 End Fraction comma plus or minus Start Fraction 2 over 7 End Fraction

plus or minus 1 comma plus or minus 2 comma plus or minus 7 comma plus or minus 14 comma plus or minus Start Fraction 7 over 2 End Fraction
Image with alt text: plus or minus 1 comma plus or minus 2 comma plus or minus 7 comma plus or minus 14 comma plus or minus Start Fraction 7 over 2 End Fraction

plus or minus 1 comma plus or minus 2 comma plus or minus 7 comma plus or minus 14
Image with alt text: plus or minus 1 comma plus or minus 2 comma plus or minus 7 comma plus or minus 14

plus or minus 1 comma plus or minus 2 comma plus or minus Start Fraction 1 over 7 End Fraction comma plus or minus Start Fraction 1 over 14 End Fraction
Image with alt text: plus or minus 1 comma plus or minus 2 comma plus or minus Start Fraction 1 over 7 End Fraction comma plus or minus Start Fraction 1 over 14 End Fraction

Answer 1

To apply the Rational Zero Theorem, we need to identify the possible rational roots of the polynomial function \( f(x) = 14x^7 - 4x^2 + 2 \).

1. **Identify the leading coefficient and the constant term**:
- Leading coefficient (the coefficient of the highest degree term): \( 14 \)
- Constant term: \( 2 \)

2. **Determine the factors of the leading coefficient (14)**:
- Factors of \( 14 \): \( \pm 1, \pm 2, \pm 7, \pm 14 \)

3. **Determine the factors of the constant term (2)**:
- Factors of \( 2 \): \( \pm 1, \pm 2 \)

4. **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:
- From \( \pm 1 \):
- \( \frac{1}{1} = \pm 1 \)
- \( \frac{1}{2} = \pm \frac{1}{2} \)
- \( \frac{1}{7} = \pm \frac{1}{7} \)
- \( \frac{1}{14} = \pm \frac{1}{14} \)

- 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)

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.