Final answer:
To factor 3z³+48z²+192z, first factor out 3z to get 3z(z² + 16z + 64), then recognize the quadratic as a perfect square trinomial (z + 8)², ending up with the expression factored completely as 3z(z + 8)².
Step-by-step explanation:
To factor the expression 3z³+48z²+192z completely, we need to look for common factors in each term. Let's start by factoring out the greatest common factor (GCF).
- First, notice that each term has a factor of 3z. So, let's factor that out: 3z(z² + 16z + 64).
- Next, we can identify that the quadratic z² + 16z + 64 is a perfect square trinomial since 64 is the square of 8 and 16 is twice the product of 8 and z. It can be factored as (z + 8)².
- Putting it all together, the fully factored form of the expression is 3z(z + 8)².