To solve the cubic equation \(-2z^3 - 22z^2 - 78z - 60 = 0\), we can follow a similar approach as before by checking for rational roots and then applying further methods if necessary.

### 1. Simplify the Equation
First, let's simplify the equation by dividing the entire equation by \(-2\) to make the coefficients smaller:
\[
z^3 + 11z^2 + 39z + 30 = 0
\]

### 2. Check for Rational Roots
Using the Rational Root Theorem, the possible rational roots are the factors of the constant term (30) divided by the factors of the leading coefficient (1). The possible rational roots are:
\[
\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30
\]

Now, let's test some of these possible roots:

- Testing \( z = -2 \):
\[
(-2)^3 + 11(-2)^2 + 39(-2) + 30 = -8 + 44 - 78 + 30 = -12 \quad (\text{not a root})
\]

- Testing

Learn how to solve the cubic equation -2z^3 - 22z^2 - 78z - 60 = 0 with a detailed step-by-step solution. This problem involves cubic equations and uses the Rational Root Theorem to find potential solutions. Suitable for high school students studying algebra and polynomial equations.

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