Subtract the polynomials. (Simplify your answer completely.) \[ \left(\frac{5}{16} s^{8}-\frac{5}{8} s^{7}\right)-\left(\frac{1}{5} s^{8}+\frac{1}{9} s^{7}\right) \]
Solution
Step 1 ：Given the polynomials \(\left(\frac{5}{16} s^{8}-\frac{5}{8} s^{7}\right)\) and \(\left(\frac{1}{5} s^{8}+\frac{1}{9} s^{7}\right)\), we are asked to subtract the second polynomial from the first.
Step 2 ：We start by identifying the coefficients of \(s^{8}\) and \(s^{7}\) in both polynomials. For \(s^{8}\), the coefficients are \(\frac{5}{16}\) and \(-\frac{1}{5}\). For \(s^{7}\), the coefficients are \(-\frac{5}{8}\) and \(-\frac{1}{9}\).
Step 3 ：We subtract the coefficients of \(s^{8}\) to get \(\frac{5}{16} - (-\frac{1}{5}) = \frac{9}{80}\). Similarly, we subtract the coefficients of \(s^{7}\) to get \(-\frac{5}{8} - (-\frac{1}{9}) = -\frac{53}{72}\).
Step 4 ：Substituting these results back into the polynomials, we get \(\frac{9}{80} s^{8} - \frac{53}{72} s^{7}\).
Step 5 ：Final Answer: \(\boxed{\frac{9}{80} s^{8} - \frac{53}{72} s^{7}}\)
