Verified Answer :

  • Rewrite the equation as \(z^3 - z^2 - 20z = 0\).
  • Factor out \(z\) to obtain \(z(z^2 - z - 20) = 0\).
  • Factor the quadratic \(z^2 - z - 20\) as \((z - 5)(z + 4)\).
  • Identify the roots \(z = 0\), \(z = 5\), and \(z = -4\).
  • The solution set is \(\boxed{0, 5, -4}\).

Explanation

  1. Standard Form of the Equation

To solve the equation \(z^3 - z^2 = 20z\), we first rewrite it in the standard form \(f(z) = 0\) by subtracting \(20z\) from both sides. This gives us:
\[z^3 - z^2 - 20z = 0\]
This is a cubic equation, and our goal is to find all values of \(z\) that satisfy this condition.

  1. Factoring out the Common Variable

We can observe that every term in the equation shares a common factor of \(z\). Factoring out \(z\) from the expression simplifies the equation:
\[z(z^2 - z - 20) = 0\]
This product is equal to zero if at least one of the factors is equal to zero. This implies two conditions:

  1. \(z = 0\)
  2. \(z^2 - z - 20 = 0\)
  1. Solving the Quadratic Equation

Now, we need to solve the quadratic equation \(z^2 - z - 20 = 0\). We look for two numbers that multiply to \(-20\) and add up to \(-1\). These numbers are \(-5\) and \(4\). Thus, we can factor the quadratic expression as follows:
\[(z - 5)(z + 4) = 0\]
Setting each factor to zero gives us the roots:

  • \(z - 5 = 0 \implies z = 5\)
  • \(z + 4 = 0 \implies z = -4\)
  1. Final Solutions

Combining the result from our first factor (\(z = 0\)) with the roots obtained from the quadratic equation (\(z = 5\) and \(z = -4\)), we conclude that the solutions for the equation are \(z = 0\), \(z = 5\), and \(z = -4\).
The final answer is \(\boxed{0, 5, -4}\).

Examples

Cubic equations often appear in geometry and physical problems. For example, if you are designing a box where the length, width, and height are related to a variable \(z\), the volume might be described by a cubic function. Solving an equation like this helps find dimensions that satisfy specific volume requirements, ensuring your structure meets the necessary specifications.

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