Polynomial Subtraction Steps

To solve the problem, we need to subtract the first expression from the second expression. This means we set up the subtraction as:

$ \left( \frac{7}{2} - \frac{x}{3} - \frac{1}{5}x^2 \right) - \left( \frac{9}{2} + \frac{x}{2} + \frac{3}{5}x^2 + \frac{7}{4}x^3 \right) $

Simplify the Expression

Distribute the negative sign to each term in the second expression:

$ \frac{7}{2} - \frac{x}{3} - \frac{1}{5}x^2 - \frac{9}{2} - \frac{x}{2} - \frac{3}{5}x^2 - \frac{7}{4}x^3 $

Combine Like Terms

Group and combine terms with the same power of $x$:

• Constant terms: $ \frac{7}{2} - \frac{9}{2} = \frac{7 - 9}{2} = \frac{-2}{2} = -1 $
• Terms with $x$: $ -\frac{x}{3} - \frac{x}{2} = x \left( -\frac{1}{3} - \frac{1}{2} \right) = x \left( \frac{-2 - 3}{6} \right) = -\frac{5}{6}x $
• Terms with $x^2$: $ -\frac{1}{5}x^2 - \frac{3}{5}x^2 = x^2 \left( -\frac{1}{5} - \frac{3}{5} \right) = x^2 \left( \frac{-1 - 3}{5} \right) = -\frac{4}{5}x^2 $
• Terms with $x^3$: $ -\frac{7}{4}x^3 $

Final Result

Combine the simplified terms to get the final expression:

$ -1 - \frac{5}{6}x - \frac{4}{5}x^2 - \frac{7}{4}x^3 $