Factor completely. If a polynomial cannot be factored, say it is prime, \( 3 y^{3}-9 y^{2}-30 y \)

To factor the polynomial \( 3y^{3} - 9y^{2} - 30y \) completely, follow these steps: 1. **Factor Out the Greatest Common Factor (GCF):** First, identify the GCF of all the terms in the polynomial. - The coefficients are 3, -9, and -30. The GCF of these numbers is **3**. - Each term contains at least one **\( y \)**. The smallest power of \( y \) present in all terms is **\( y \)**. So, the GCF is **\( 3y \)**. Factor out the GCF: \[ 3y^{3} - 9y^{2} - 30y = 3y(y^{2} - 3y - 10) \] 2. **Factor the Quadratic Expression:** Now, focus on factoring the quadratic \( y^{2} - 3y - 10 \). We need to find two numbers that: - Multiply to **\(-10\)** (the constant term). - Add up to **\(-3\)** (the coefficient of the middle term). The numbers **\(-5\)** and **\(2\)** satisfy these conditions: \[ (-5) \times 2 = -10 \\ (-5) + 2 = -3 \] So, the quadratic factors as: \[ y^{2} - 3y - 10 = (y - 5)(y + 2) \] 3. **Combine the Factors:** Put it all together: \[ 3y^{3} - 9y^{2} - 30y = 3y(y - 5)(y + 2) \] **Final Factored Form:** \[ 3y(y - 5)(y + 2) \]