List the possible rational zeros. $$ n(x)=-4 x^{6}-16 x^{3}-7 x-6 $$ If there is more than one answer, separate them with commas. Select "None" if applicable. The possible rational zeros of $$ n(x): \square $$

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UpStudy AI · Accepted Answer

We first apply the Rational Root Theorem. For a polynomial  
$$
n(x)=-4x^6-16x^3-7x-6,
$$  
the possible rational zeros are of the form  
$$
\frac{p}{q},
$$  
where $$p$$ divides the constant term $$-6$$ and $$q$$ divides the leading coefficient $$-4$$.

**Step 1. List the factors.**

- Factors of the constant term $$-6$$:  
  $$
  \pm 1,\, \pm 2,\, \pm 3,\, \pm 6.
  $$

- Factors of the leading coefficient $$-4$$:  
  $$
  \pm 1,\, \pm 2,\, \pm 4.
  $$

**Step 2. Write the possible rational zeros $$\frac{p}{q}$$.**

The possible values of $$ \frac{p}{q} $$ are:
- Using $$q=1$$:  
  $$\pm 1,\, \pm 2,\, \pm 3,\, \pm 6$$.

- Using $$q=2$$:  
  $$\pm \frac{1}{2},\, \pm \frac{2}{2} \,(\text{which is} \, \pm 1),\, \pm \frac{3}{2},\, \pm \frac{6}{2} \,(\text{which is} \, \pm 3)$$.

- Using $$q=4$$:  
  $$\pm \frac{1}{4},\, \pm \frac{2}{4} \,(\text{which is} \, \pm \frac{1}{2}),\, \pm \frac{3}{4},\, \pm \frac{6}{4} \,(\text{which is} \, \pm \frac{3}{2})$$.

**Step 3. Combine and remove duplicates.**

Thus, the unique possible rational zeros are:  
$$
\pm 1, \, \pm 2, \, \pm 3, \, \pm 6, \, \pm \frac{1}{2}, \, \pm \frac{3}{2}, \, \pm \frac{1}{4}, \, \pm \frac{3}{4}.
$$

**Final Answer:**

The possible rational zeros of $$ n(x) $$ are  
$$
\pm 1,\, \pm 2,\, \pm 3,\, \pm 6,\, \pm \frac{1}{2},\, \pm \frac{3}{2},\, \pm \frac{1}{4},\, \pm \frac{3}{4}.
$$
The possible rational zeros of $$ n(x) $$ are $$ \pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{1}{4}, \pm\frac{3}{4} $$.

List the possible rational zeros. \[ n(x)=-4 x^{6}-16 x^{3}-7 x-6 \] If there is more than one answer, separate them with commas. Select "None" if applicable. The possible rational zeros of \( n(x): \square \)

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