Question

Use the Rational Zero Theorem to list all possible rational zeros for the given function.
​f(x)equals x cubed minus 6 x squared minus 16 x plus 14
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Part 1
Choose the answer below that lists all possible rational zeros.
A.negative 1 comma 1 comma negative 2 comma 2 comma negative 7 comma 7 comma negative 14 comma 14 comma negative one half comma one half comma negative one seventh comma one seventh comma negative one fourteenth comma one fourteenth
negative 1 comma 1 comma negative 2 comma 2 comma negative 7 comma 7 comma negative 14 comma 14 comma negative one half comma one half comma negative one seventh comma one seventh comma negative one fourteenth comma one fourteenth
B.negative 1 comma 1 comma negative one half comma one half comma negative one seventh comma one seventh comma negative one fourteenth comma one fourteenth
negative 1 comma 1 comma negative one half comma one half comma negative one seventh comma one seventh comma negative one fourteenth comma one fourteenth
C.negative 1 comma 1 comma negative 14 comma 14
negative 1 comma 1 comma negative 14 comma 14
D.negative 1 comma 1 comma negative 2 comma 2 comma negative 7 comma 7 comma negative 14 comma 14
negative 1 comma 1 comma negative 2 comma 2 comma negative 7 comma 7 comma negative 14 comma 14

Answer 1

To use the Rational Zero Theorem to find the possible rational zeros of the polynomial \(f(x) = x^3 - 6x^2 - 16x + 14\), we need to consider the factors of the constant term and the leading coefficient.

1. The constant term is \(14\).
2. The factors of \(14\) are:
- Positive factors: \(1, 2, 7, 14\)
- Negative factors: \(-1, -2, -7, -14\)

So all factors of \(14\) are:
\[
\pm 1, \pm 2, \pm 7, \pm 14
\]

3. The leading coefficient (the coefficient of \(x^3\)) is \(1\), and its factors are:
- Positive factors: \(1\)
- Negative factors: \(-1\)

The only factors of \(1\) are:
\[
\pm 1
\]

According to the Rational Zeros Theorem, the possible rational zeros are the ratios of the factors of the constant term to the factors of the leading coefficient. Therefore, the possible rational zeros of the polynomial are:

\[
\pm 1, \pm 2, \pm 7, \pm 14
\]

There are no other values because there are no factors of the leading coefficient other than \(1\).

Now, looking at the provided options:

- A: \( -1, 1, -2, 2, -7, 7, -14, 14, -\frac{1}{2}, \frac{1}{2}, -\frac{1}{7}, \frac{1}{7}, -\frac{1}{14}, \frac{1}{14} \)
- B: \( -1, 1, -\frac{1}{2}, \frac{1}{2}, -\frac{1}{7}, \frac{1}{7}, -\frac{1}{14}, \frac{1}{14} \)
- C: \( -1, 1, -14, 14 \)
- D: \( -1, 1, -2, 2, -7, 7, -14, 14 \)

Based on our analysis, option D (\(-1, 1, -2, 2, -7, 7, -14, 14\)) lists all the correct possible rational zeros for the given function.

**Answer: D.**