List the possible rational zeros of $$ f(x)=2 x^{4}-3 x^{3}+19 x^{2}-27 x+9 $$ How many possible zeros does the following polynomial have?

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To find the possible rational zeros of the polynomial $$ f(x) = 2x^4 - 3x^3 + 19x^2 - 27x + 9 $$, we can use the Rational Root Theorem. This theorem states that any rational solution, expressed as a fraction $$ \frac{p}{q} $$, must have $$ p $$ as a factor of the constant term and $$ q $$ as a factor of the leading coefficient.

1. **Identify the constant term and leading coefficient:**
   - The constant term $$ a_0 = 9 $$.
   - The leading coefficient $$ a_n = 2 $$.

2. **Find the factors of the constant term (9):**
   - The factors of 9 are $$ \pm 1, \pm 3, \pm 9 $$.

3. **Find the factors of the leading coefficient (2):**
   - The factors of 2 are $$ \pm 1, \pm 2 $$.

4. **List the possible rational zeros:**
   - The possible rational zeros are given by $$ \frac{p}{q} $$, where $$ p $$ is a factor of 9 and $$ q $$ is a factor of 2.
   - This gives us the following combinations:
     - $$ \frac{1}{1} = 1 $$
     - $$ \frac{1}{2} = \frac{1}{2} $$
     - $$ \frac{3}{1} = 3 $$
     - $$ \frac{3}{2} = \frac{3}{2} $$
     - $$ \frac{9}{1} = 9 $$
     - $$ \frac{9}{2} = \frac{9}{2} $$
     - $$ \frac{-1}{1} = -1 $$
     - $$ \frac{-1}{2} = -\frac{1}{2} $$
     - $$ \frac{-3}{1} = -3 $$
     - $$ \frac{-3}{2} = -\frac{3}{2} $$
     - $$ \frac{-9}{1} = -9 $$
     - $$ \frac{-9}{2} = -\frac{9}{2} $$

5. **Summarize the possible rational zeros:**
   - The possible rational zeros are:
     $$
     \pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 9, \pm \frac{9}{2}
     $$

6. **Count the number of possible rational zeros:**
   - There are a total of 12 possible rational zeros.

Thus, the possible rational zeros of the polynomial $$ f(x) $$ are $$ \pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 9, \pm \frac{9}{2} $$, and there are 12 possible rational zeros in total.
The possible rational zeros of the polynomial $$ f(x) = 2x^4 - 3x^3 + 19x^2 - 27x + 9 $$ are $$ \pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 9, \pm \frac{9}{2} $$. There are 12 possible rational zeros in total.

List the possible rational zeros of How many possible zeros does the following polynomial have?

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