To simplify the expression $3x^3 - 6x^2 - 24x$, we can factor out the greatest common factor.

Step 1: Factor out the common term

The common factor in each term is $3x$: $3x^3 - 6x^2 - 24x = 3x(x^2 - 2x - 8)$

Step 2: Factor the quadratic expression

Now we focus on $x^2 - 2x - 8$. We need to find two numbers that multiply to $-8$ and add up to $-2$. These numbers are $-4$ and $2$.

Thus, we can factor $x^2 - 2x - 8$ as: $x^2 - 2x - 8 = (x - 4)(x + 2)$

Step 3: Combine the factors

Now we substitute back into the original expression: $3x^3 - 6x^2 - 24x = 3x(x - 4)(x + 2)$

Final Answer

$3x^3 - 6x^2 - 24x = 3x(x - 4)(x + 2)$

Would you like more details on factoring or have any questions?

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1. How do you identify the greatest common factor in polynomials?
2. What are the steps to factor a quadratic expression?
3. How does factoring help in solving polynomial equations?
4. Can this factored form be used to find roots of the polynomial?
5. How can you verify that the factorization is correct?

Tip: Always check your factored form by expanding to ensure accuracy.