Simplify the following:
(z^2-25/z^2) (z^2+2 z-35-(5 z)/z^2)
Combine powers. z/z^2 = z^(1-2):
(z^2-25/z^2) (z^2+2 z-35-5 z^1-2)
1-2 = -1:
(z^2-25/z^2) (z^2+2 z-35-5 z^-1)
Put each term in z^2+2 z-35-5/z over the common denominator z: z^2+2 z-35-5/z = z^3/z+(2 z^2)/z-(35 z)/z-5/z:
z^3/z+(2 z^2)/z-(35 z)/z-5/z (z^2-25/z^2)
z^3/z+(2 z^2)/z-(35 z)/z-5/z = (z^3+2 z^2-35 z-5)/z:
(z^3+2 z^2-35 z-5)/z (z^2-25/z^2)
Put each term in z^2-25/z^2 over the common denominator z^2: z^2-25/z^2 = z^4/z^2-25/z^2:
(z^4/z^2-25/z^2 (z^3+2 z^2-35 z-5))/(z)
z^4/z^2-25/z^2 = (z^4-25)/z^2:
((z^4-25)/z^2 (z^3+2 z^2-35 z-5))/(z)
z^4-25 = (z^2)^2-5^2:
((z^2)^2-5^2 (z^3+2 z^2-35 z-5))/(z^2 z)
Factor the difference of two squares. (z^2)^2-5^2 = (z^2-5) (z^2+5):
((z^2-5) (z^2+5) (z^3+2 z^2-35 z-5))/(z^2 z)
Combine powers. ((z^2-5) (z^2+5) (z^3+2 z^2-35 z-5))/(z^2 z) = z^(-2-1) (z^2-5) (z^2+5) (z^3+2 z^2-35 z-5):
z^(-2-1) (z^2-5) (z^2+5) (z^3+2 z^2-35 z-5)
-2-1 = -3:
Answer: |
| z^-3 (z^2-5) (z^2+5) (z^3+2z^2-35z-5)
