To factorize the polynomial $3x^3 + 17x^2 + 22x + 8$ using the Factor Theorem, we need to find at least one root of the polynomial. If we find a root $r$, then $x - r$ is a factor of the polynomial.

Let's go through the steps:

1. Identify Possible Rational Roots: According to the Rational Root Theorem, any rational root of the polynomial must be a factor of the constant term (8) divided by a factor of the leading coefficient (3). So, the possible rational roots are: $\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3}$

2. Testing Possible Roots: We substitute these values into the polynomial to see if any of them yield zero. Starting with simpler values:

   • For $x = -1$: $3(-1)^3 + 17(-1)^2 + 22(-1) + 8 = -3 + 17 - 22 + 8 = 0$ Since $f(-1) = 0$, $x = -1$ is a root, and therefore $x + 1$ is a factor.

3. Divide the Polynomial by $x + 1$: Now, we perform synthetic division or polynomial division to divide $3x^3 + 17x^2 + 22x + 8$ by $x + 1$.

   After dividing, we get: $3x^3 + 17x^2 + 22x + 8 = (x + 1)(3x^2 + 14x + 8)$

4. Factor the Quotient Further: Now, we need to factor $3x^2 + 14x + 8$. We can use factoring techniques or the quadratic formula to find the factors.

   Solving $3x^2 + 14x + 8 = 0$ gives the roots: $x = -\frac{4}{3} \text{ and } x = -2$ So, $3x^2 + 14x + 8 = (3x + 4)(x + 2)$.

5. Write the Final Factorization: Combining all factors, we get: $3x^3 + 17x^2 + 22x + 8 = (x + 1)(3x + 4)(x + 2)$

So, the factorization of $3x^3 + 17x^2 + 22x + 8$ is: $(x + 1)(3x + 4)(x + 2)$

Would you like further details on any of the steps?

Here are some related questions that might be useful:

1. How does the Factor Theorem help in factorizing polynomials?
2. Can synthetic division be used for polynomials with non-integer roots?
3. What is the Rational Root Theorem, and how does it work?
4. How do you determine the possible rational roots of a polynomial?
5. What are other methods to factorize a polynomial if the Factor Theorem does not apply?

Tip: Always verify your factors by multiplying them to see if you get the original polynomial. This ensures accuracy in factorization.