To factor the polynomial \(3x^3 - 6x^2 - 72x\), we can start by factoring out the greatest common factor (GCF) from the terms.
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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 \(x\).
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Factor out the GCF: \[ 3x^3 - 6x^2 - 72x = 3x(x^2 - 2x - 24) \]
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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: \[ (-6) \times 4 = -24 \quad \text{and} \quad (-6) + 4 = -2 \]
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So we can factor \(x^2 - 2x - 24\) as: \[ x^2 - 2x - 24 = (x - 6)(x + 4) \]
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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: \(3x(x+4)(x−6)\).
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