The given polynomial is:

$3x^3 - 4x^2 + 8x + 8$

Here’s a detailed breakdown of potential analyses and computations for this polynomial:

1. Factorization

To attempt factorization, we can use synthetic division or test possible roots using the Rational Root Theorem. The Rational Root Theorem suggests that any rational root is of the form:

$\pm \frac{\text{factor of constant term}}{\text{factor of leading coefficient}}$

Here:

• Constant term: $8$
• Leading coefficient: $3$

Possible roots: $\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3}$

Testing roots like $x = -2$ shows it is a root, so $x + 2$ is a factor.

2. Derivative (for critical points or slope analysis)

The derivative of $3x^3 - 4x^2 + 8x + 8$ is: $f'(x) = 9x^2 - 8x + 8$

This can help find the critical points or intervals of increase/decrease.

3. Roots (if any)

After identifying one factor, the remaining quadratic can be solved using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

4. Graphical Behavior

• As a cubic function, the polynomial has at least one real root.
• Leading term $3x^3$: As $x \to \pm \infty$, $f(x) \to \pm \infty$.

5. Value at Specific Points

Plug in values for $x$ to analyze behavior at particular points (e.g., $f(0) = 8$).

Would you like me to:

• Factorize completely?
• Solve for all roots?
• Provide a graph of this polynomial?
• Find the critical points and intervals of increase/decrease?

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Relative Questions:

1. How do you determine possible rational roots of a polynomial?
2. What methods are effective for factorizing cubic polynomials?
3. How does the derivative inform the behavior of a cubic polynomial?
4. Why does a cubic function always have at least one real root?
5. What are some applications of cubic polynomials in real-world contexts?

Tip:

When analyzing polynomials, always start by testing simple roots (like $\pm 1, \pm 2$) to simplify factorization.