Question

factor ( z^3 - 4z^2 - 5z ) completely.

Explanation:

Step1: Factor out the GCF

The greatest common factor (GCF) of \(z^3\), \(-4z^2\), and \(-5z\) is \(z\). So we factor out \(z\) from each term:
\(z^3 - 4z^2 - 5z = z(z^2 - 4z - 5)\)

Step2: Factor the quadratic

Now we need to factor the quadratic \(z^2 - 4z - 5\). We look for two numbers that multiply to \(-5\) and add up to \(-4\). The numbers \(-5\) and \(1\) work because \(-5 \times 1 = -5\) and \(-5 + 1 = -4\). So we can factor the quadratic as:
\(z^2 - 4z - 5 = (z - 5)(z + 1)\)

Step3: Combine the factors

Putting it all together, the completely factored form of the original expression is:
\(z(z - 5)(z + 1)\)

Answer:

\(z(z - 5)(z + 1)\)