Question

What are the linear factors of the polynomial 3x3−6x2−72x ? Option #1: 3x(x+4)(x−6) Option #2: x(x+4)(x−6) Option #3: 3(x+4)(x−6)

Answer 1

To factor the polynomial \(3x^3 - 6x^2 - 72x\), we can start by factoring out the greatest common factor (GCF) from the terms.

1. Identify the GCF:
- The coefficients are 3, -6, and -72. The GCF of these numbers is 3.
- Each term has at least one factor of \(x\).

2. Factor out the GCF:
\[
3x^3 - 6x^2 - 72x = 3x(x^2 - 2x - 24)
\]

3. Now, we need to factor the quadratic expression \(x^2 - 2x - 24\). We look for two numbers that multiply to \(-24\) and add to \(-2\). The numbers \(-6\) and \(4\) work, as:
\[
(-6) \times 4 = -24 \quad \text{and} \quad (-6) + 4 = -2
\]

4. So we can factor \(x^2 - 2x - 24\) as:
\[
x^2 - 2x - 24 = (x - 6)(x + 4)
\]

5. Combining this with our earlier factor, we have:
\[
3x^3 - 6x^2 - 72x = 3x(x - 6)(x + 4)
\]

Now we compare this with the options provided:

- **Option #1**: \(3x(x+4)(x−6)\) - This does match with our result after adjusting the order.
- **Option #2**: \(x(x+4)(x−6)\) - This doesn't match because it lacks the factor of 3.
- **Option #3**: \(3(x+4)(x−6)\) - This doesn't match the factorization, as it lacks the factor of \(x\).

Thus, the correct answer is **Option #1: \(3x(x+4)(x−6)\)**.