Find the zeros of the polynomial p(x) = 2x
3 + 3x
2 − 18x − 27.

To find the zeros of the polynomial \( p(x) = 2x^3 + 3x^2 - 18x - 27 \), we need to solve the equation \( p(x) = 0 \).

### Step 1: Apply the Rational Root Theorem
The Rational Root Theorem states that any possible rational root of a polynomial with integer coefficients is a fraction \(\frac{p}{q}\), where:
- \(p\) is a factor of the constant term (in this case, \(-27\)),
- \(q\) is a factor of the leading coefficient (in this case, \(2\)).

The factors of \(-27\) are \(\pm 1, \pm 3, \pm 9, \pm 27\).

The factors of \(2\) are \(\pm 1, \pm 2\).

Thus, the possible rational roots are:
\[
\pm 1, \pm 3, \pm 9, \pm 27, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2}, \pm \frac{27}{2}
\]

### Step 2: Test the Possible Rational Roots
We substitute each possible root into the polynomial to check if it equals zero.

Let's try \(x = -3\):

\[
p(-3) = 2(-3)^3 + 3(-3)^2 - 18(-3) - 27
\]

Calculating each term:

\[
p(-3) = 2(-27) + 3(9) + 54 - 27
\]

\[
p(-3) = -54 + 27 + 54 - 27 = 0
\]

So, \(x = -3\) is a root of the polynomial.

### Step 3: Perform Polynomial Division
Since \(x = -3\) is a root, \(x + 3\) is a factor of \(p(x)\). We can use synthetic division or polynomial long division to divide \(p(x)\) by \(x + 3\).

Let's use synthetic division:

\[
\begin{array}{r|rrrr}
-3 & 2 & 3 & -18 & -27 \\
   &   & -6 & 9  & 27  \\
\hline
   & 2 & -3 & -9 & 0 \\
\end{array}
\]

The quotient is \(2x^2 - 3x - 9\).

### Step 4: Factor the Quotient
Now, we factor the quadratic \(2x^2 - 3x - 9 = 0\).

To factor this quadratic, we look for two numbers that multiply to \(2 \cdot (-9) = -18\) and add to \(-3\).

These numbers are \(-6\) and \(3\). So, we rewrite the quadratic as:

\[
2x^2 - 6x + 3x - 9 = 0
\]

Grouping the terms:

\[
2x(x - 3) + 3(x - 3) = 0
\]

Factoring by grouping:

\[
(x - 3)(2x + 3) = 0
\]

### Step 5: Solve for the Remaining Roots
Now set each factor equal to zero:

1. \(x - 3 = 0 \implies x = 3\)
2. \(2x + 3 = 0 \implies x = -\frac{3}{2}\)

### Step 6: Conclusion
The zeros of the polynomial \(p(x) = 2x^3 + 3x^2 - 18x - 27\) are:

\[
x = -3, \quad x = 3, \quad x = -\frac{3}{2}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you use synthetic division to divide polynomials?
2. What is the Rational Root Theorem and how does it work?
3. How do you factor a quadratic expression?
4. How do you solve a cubic equation using factoring?
5. What are the different methods for finding the roots of polynomials?

### Tip:
When dealing with polynomials, always check for possible rational roots first to simplify the solving process.

Learn how to find the zeros of the polynomial 2x^3 + 3x^2 - 18x - 27 with a step-by-step solution. This problem involves applying the Rational Root Theorem, synthetic division, and factoring quadratic expressions. Suitable for students in Grades 9-12.

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