Find the simplified quotient. \( \frac{t^{2}-36}{3} \div \frac{t^{2}+E t}{9 t} \) \( \frac{t-6}{3} \) \( t-6 \) \( 3 t+18 \) \( 3 t-18 \) DONE

To simplify the expression \( \frac{t^{2}-36}{3} \div \frac{t^{2}+E t}{9 t} \), start by rewriting the division as multiplication by the reciprocal: \[ = \frac{t^{2}-36}{3} \times \frac{9t}{t^{2}+Et} \] Now, factor \( t^{2} - 36 \) as \( (t - 6)(t + 6) \) and simplify: \[ = \frac{(t-6)(t+6)}{3} \times \frac{9t}{t^{2}+Et} \] Depending on the value of \( E \), you may further factor or simplify the second part, but we can't proceed without knowing \( E \). If you substitute a specific value for \( E \) later, you can determine the final simplified quotient. Real-world scenarios often involve polynomial expressions similar to this. For instance, engineers might simplify such expressions when calculating trajectories or designing curved structures, ensuring they balance forces efficiently. Mathematicians also take special joy in polynomial factorization. The beauty of expressions like \( t^2 - 36 \) is their connections to fundamental algebraic identities, like the difference of squares. Whether you’re working on geometry or calculus, recognizing these patterns can turn a headache of a function into an elegantly solved equation!