To find all zeros of the polynomial $f(x) = x^3 + 2x^2 - 19x + 30$, we can approach this by factoring the polynomial and applying the Linear Factor Theorem. Here’s how you can solve it step-by-step:

1. Check for Rational Roots: Use the Rational Root Theorem to test possible rational roots, which would be the factors of the constant term (30) divided by the factors of the leading coefficient (1). So, the possible rational roots are $\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30$.

2. Test Roots: Substitute each possible root into $f(x)$ to see if it equals zero. For example:

   • $f(1) = 1^3 + 2(1)^2 - 19(1) + 30 = 1 + 2 - 19 + 30 = 14$ (not zero)
   • $f(-1) = (-1)^3 + 2(-1)^2 - 19(-1) + 30 = -1 + 2 + 19 + 30 = 50$ (not zero)
   • Continue this process for each possible root until you find one that makes $f(x) = 0$.

3. Factor Polynomial: Once you find a root $r$, say $x = a$, you can then divide $f(x)$ by $x - a$ to reduce the polynomial’s degree. Repeat this until you have factored the polynomial completely.

4. Solve for Remaining Factors: After factoring, set each factor equal to zero to solve for all roots, including multiplicities if any factor repeats.

Let me know if you'd like a detailed step-by-step solution for each potential root test or further guidance on polynomial division!

Would you like a breakdown of each step, or do you have questions about specific parts of this process?

Here are some related questions:

1. How does the Rational Root Theorem help in finding zeros of a polynomial?
2. What is the significance of the Linear Factor Theorem in polynomial factorization?
3. How do we handle polynomials with higher degrees when finding zeros?
4. Why do we consider multiplicity when listing solutions for polynomial roots?
5. How does synthetic division work in simplifying polynomial division?

Tip: Checking for rational roots can save time and simplify finding solutions for higher-degree polynomials by narrowing down possible roots.