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The problem in the image asks to "Find possible rational roots, rational root of the polynomial" and gives the polynomial:

\[
x^4 - 3x^3 - 3x^2 + 3x + 8 = 0
\]

To find the possible rational roots of this polynomial, we use the Rational Root Theorem. The theorem states that any rational root, expressed as a fraction \(\frac{p}{q}\), will have \(p\) as a factor of the constant term (in this case, 8) and \(q\) as a factor of the leading coefficient (in this case, 1).

### Step 1: Identify factors
- The constant term is 8, so the factors of 8 are: \( \pm 1, \pm 2, \pm 4, \pm 8 \).
- The leading coefficient is 1, so the factors of 1 are: \( \pm 1 \).

### Step 2: List possible rational roots
Given these factors, the possible rational roots (using \( \frac{p}{q} \)) are:
\[
\pm 1, \pm 2, \pm 4, \pm 8
\]

### Step 3: Test each possible root
To check which of these are actual roots, we substitute each one into the polynomial equation:

- For \(x = 1\):
  \[
  1^4 - 3(1)^3 - 3(1)^2 + 3(1) + 8 = 1 - 3 - 3 + 3 + 8 = 6 \quad (\text{not a root})
  \]
- For \(x = -1\):
  \[
  (-1)^4 - 3(-1)^3 - 3(-1)^2 + 3(-1) + 8 = 1 + 3 - 3 - 3 + 8 = 6 \quad (\text{not a root})
  \]
- For \(x = 2\):
  \[
  2^4 - 3(2)^3 - 3(2)^2 + 3(2) + 8 = 16 - 24 - 12 + 6 + 8 = -6 \quad (\text{not a root})
  \]
- For \(x = -2\):
  \[
  (-2)^4 - 3(-2)^3 - 3(-2)^2 + 3(-2) + 8 = 16 + 24 - 12 - 6 + 8 = 30 \quad (\text{not a root})
  \]
- For \(x = 4\):
  \[
  4^4 - 3(4)^3 - 3(4)^2 + 3(4) + 8 = 256 - 192 - 48 + 12 + 8 = 36 \quad (\text{not a root})
  \]
- For \(x = -4\):
  \[
  (-4)^4 - 3(-4)^3 - 3(-4)^2 + 3(-4) + 8 = 256 + 192 - 48 - 12 + 8 = 396 \quad (\text{not a root})
  \]
- For \(x = 8\):
  \[
  8^4 - 3(8)^3 - 3(8)^2 + 3(8) + 8 = 4096 - 1536 - 192 + 24 + 8 = 2400 \quad (\text{not a root})
  \]
- For \(x = -8\):
  \[
  (-8)^4 - 3(-8)^3 - 3(-8)^2 + 3(-8) + 8 = 4096 + 1536 - 192 - 24 + 8 = 5424 \quad (\text{not a root})
  \]

None of the possible rational roots seem to satisfy the polynomial equation, meaning that the polynomial might not have rational roots, or the roots might be irrational or complex.

Would you like more details or have any questions? Here are some related topics we can explore:

1. What is the process for factoring polynomials with irrational or complex roots?
2. How can the Rational Root Theorem be extended to higher-degree polynomials?
3. Can this polynomial be factored using numerical methods or approximations?
4. What are synthetic division and how is it used to test roots?
5. How do complex numbers play a role in finding roots of polynomials?

**Tip:** When solving polynomials, after using the Rational Root Theorem, considering numerical methods or graphing tools can help identify roots that might not be rational.

Find possible rational roots, rational root of the polynomial: x^4 - 3x^3 - 3x^2 + 3x + 8 = 0

Explore the process of finding possible rational roots for the polynomial equation x^4 - 3x^3 - 3x^2 + 3x + 8 = 0 using the Rational Root Theorem. This problem is ideal for students in Grades 10-12, focusing on key algebraic concepts and polynomial root-finding techniques.

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