Question

What are the linear factors of the polynomial 3x3−6x2−72x%0D%0A3%0D%0A𝑥%0D%0A3%0D%0A−%0D%0A6%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A72%0D%0A𝑥%0D%0A?%0D%0A%0D%0AOption #1: 3x(x+4)(x−6)%0D%0A3%0D%0A𝑥%0D%0A(%0D%0A𝑥%0D%0A+%0D%0A4%0D%0A)%0D%0A(%0D%0A𝑥%0D%0A−%0D%0A6%0D%0A)%0D%0A%0D%0AOption #2: x(x+4)(x−6)%0D%0A𝑥%0D%0A(%0D%0A𝑥%0D%0A+%0D%0A4%0D%0A)%0D%0A(%0D%0A𝑥%0D%0A−%0D%0A6%0D%0A)%0D%0A%0D%0AOption #3: 3(x+4)(x−6)

Answer 1

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

1. **Identify the GCF**: The coefficients are 3, -6, and -72. The GCF of these coefficients is 3. The variable factor is \( x \) since all terms contain \( x \).

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

3. **Factor the quadratic \( x^2 - 2x - 24 \)**: We are looking for two numbers that multiply to \(-24\) (the constant term) and add up to \(-2\) (the coefficient of \( x \)). The numbers \(-6\) and \(4\) work because \(-6 \cdot 4 = -24\) and \(-6 + 4 = -2\).

4. **Factor the quadratic**:
\[
x^2 - 2x - 24 = (x - 6)(x + 4)
\]

5. **Putting it all together**:
\[
3x^3 - 6x^2 - 72x = 3x(x - 6)(x + 4)
\]

Thus, the linear factors of the polynomial \( 3x^3 - 6x^2 - 72x \) are:

\[
3x(x + 4)(x - 6)
\]

Therefore, the correct answer from the given options is:

**Option #1: \( 3x(x + 4)(x - 6) \)**.