To solve the polynomial equation \( 2x^4 + 3x^3 - 3x - 2 = 0 \), we can start by looking for rational roots using the Rational Root Theorem. The Rational Root Theorem states that any rational solution, in the form of \( \frac{p}{q} \), is a factor of the constant term (\(-2\)) divided by a factor of the leading coefficient (\(2\)).

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

Possible rational roots are \( \pm 1, \pm 2, \pm \frac{1}{2} \).

We test these possible roots by substituting them into the polynomial:

1. Testing \( x = 1 \):
   \[
   2(1)^4 + 3(1)^3 - 3(1) - 2 = 2 + 3 - 3 - 2 = 0
   \]
   \( x = 1 \) is a root.

Next, we use polynomial division or synthetic division to factor out \( x - 1 \) from \( 2x^4 + 3x^3 - 3x - 2 \).

Performing synthetic division with \( x = 1 \):

```
1 |  2   3   0  -3  -2
   |      2   5   5   2
   -----------------
     2   5   5   2   0
```

The quotient is \( 2x^3 + 5x^2 + 5x + 2 \).

So, the polynomial factors as:
\[
2x^4 + 3x^3 - 3x - 2 = (x - 1)(2x^3 + 5x^2 + 5x + 2)
\]

Next, we find the roots of the cubic polynomial \( 2x^3 + 5x^2 + 5x + 2 \).

We use the Rational Root Theorem again for \( 2x^3 + 5x^2 + 5x + 2 \):
Possible rational roots are \( \pm 1, \pm 2, \pm \frac{1}{2} \).

Testing \( x = -1 \):
\[
2(-1)^3 + 5(-1)^2 + 5(-1) + 2 = -2 + 5 - 5 + 2 = 0
\]
\( x = -1 \) is a root.

Performing synthetic division with \( x = -1 \):

```
-1 |  2   5   5   2
    |     -2  -3  -2
    ---------------
      2   3   2   0
```

The quotient is \( 2x^2 + 3x + 2 \).

So, we have:
\[
2x^3 + 5x^2 + 5x + 2 = (x + 1)(2x^2 + 3x + 2)
\]

Now, factor \( 2x^2 + 3x + 2 \):
We look for factors that multiply to \(2 \cdot 2 = 4\) and add to \(3\).

The factors \(1\) and \(2\) work:
\[
2x^2 + 3x + 2 = (2x + 1)(x + 2)
\]

Putting it all together, we have:
\[
2x^4 + 3x^3 - 3x - 2 = (x - 1)(x + 1)(2x + 1)(x + 2)
\]

So, the roots of the polynomial are:
\[
x = 1, -1, -\frac{1}{2}, -2
\]

Would you like further details or have any questions?

Here are 8 related questions to expand your understanding:
1. How does the Rational Root Theorem help in finding polynomial roots?
2. Can you explain the process of synthetic division?
3. How would you solve a polynomial of degree 5?
4. What are some alternative methods to solve higher degree polynomials?
5. How can we verify the roots of a polynomial?
6. What are complex roots and how are they found?
7. Can a polynomial have more roots than its degree?
8. How do multiplicities of roots affect the polynomial graph?

**Tip:** Always check your roots by substituting them back into the original polynomial to ensure they satisfy the equation.

Learn how to solve the polynomial equation 2x^4 + 3x^3 - 3x - 2 with a detailed step-by-step solution. This problem covers polynomial equations, the Rational Root Theorem, and synthetic division. Suitable for students in Grades 10-12.

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