\( 1 \leftarrow \quad \begin{array}{l}\frac{40 x^{7}-45 x^{4}+25 x}{5 x} \\ \begin{array}{l}\text { Divide and check. } \\ \text { The answer is } \\ \text { (Simplify your answer.) }\end{array}\end{array} \)

To simplify the expression \( \frac{40 x^{7}-45 x^{4}+25 x}{5 x} \), you start by factoring out the common term in the numerator. Since each term in the numerator includes \( x \), you can rewrite it as: \[ \frac{5x(8x^6 - 9x^3 + 5)}{5x} \] Now, when you divide by \( 5x \), the \( 5x \) in the numerator and the \( 5x \) in the denominator cancel each other out, leaving you with: \[ 8x^6 - 9x^3 + 5 \] That’s your simplified answer! For a real-world application, this type of algebraic simplification is often used in engineering and physics, where manipulation of polynomial expressions is essential. For example, when determining the motion of objects under gravity, one often encounters equations representing trajectories, where simplifying these can lead to clearer insights and solutions. When simplifying expressions, a common mistake is forgetting to cancel out all like terms or misapplying the laws of exponents. Always double-check your work by substituting a value for \( x \) into both the original and simplified expressions to ensure they're equivalent. This technique can save you from overlooking errors!