Divide. \[ \frac{9 v^{7}+24 v^{5}-6 v^{4}}{3 v^{3}} \]

To divide the expression \( \frac{9 v^{7}+24 v^{5}-6 v^{4}}{3 v^{3}} \), we can separate the terms in the numerator and divide each one by the denominator: \[ \frac{9 v^{7}}{3 v^{3}} + \frac{24 v^{5}}{3 v^{3}} - \frac{6 v^{4}}{3 v^{3}} \] Now, let's simplify each term: 1. \( \frac{9 v^{7}}{3 v^{3}} = 3 v^{4} \) (since \( 9 \div 3 = 3 \) and \( v^{7} \div v^{3} = v^{4} \)). 2. \( \frac{24 v^{5}}{3 v^{3}} = 8 v^{2} \) (because \( 24 \div 3 = 8 \) and \( v^{5} \div v^{3} = v^{2} \)). 3. \( \frac{6 v^{4}}{3 v^{3}} = 2 v \) (for \( 6 \div 3 = 2 \) and \( v^{4} \div v^{3} = v \)). Putting it all together, we get: \[ 3 v^{4} + 8 v^{2} - 2 v \] So, the final simplified form is: \[ 3 v^{4} + 8 v^{2} - 2 v \]