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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)

1 answer

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: \( 3x(x + 4)(x - 6) \).