Question

What is the difference of the polynomials? (8r^6s^3 – 9r^5s^4 + 3r^4s^5) – (2r^4s^5 – 5r^3s^6 – 4r^5s^4)

Answer 1

Simplify the following:
8 r^6 s^3 - 9 r^5 s^4 + 3 r^4 s^5 - (2 r^4 s^5 - 5 r^3 s^6 - 4 r^5 s^4)
Factor r^3 s^4 out of 2 r^4 s^5 - 5 r^3 s^6 - 4 r^5 s^4:
8 r^6 s^3 - 9 r^5 s^4 + 3 r^4 s^5 - r^3 s^4 (2 r s - 5 s^2 - 4 r^2)
Factor r^3 out of 8 r^6 s^3 - 9 r^5 s^4 + 3 r^4 s^5 - r^3 s^4 (2 r s - 5 s^2 - 4 r^2),
resulting in r^3 (8 s^3 r^(6 - 3) - 9 s^4 r^(5 - 3) + 3 s^5 r^(4 - 3) - s^4 (2 r s - 5 s^2 - 4 r^2)):
r^3 (8 s^3 r^(6 - 3) - 9 s^4 r^(5 - 3) + 3 s^5 r^(4 - 3) - s^4 (2 r s - 5 s^2 - 4 r^2))
6 - 3 = 3:
r^3 (8 s^3 r^3 - 9 s^4 r^(5 - 3) + 3 s^5 r^(4 - 3) - s^4 (2 r s - 5 s^2 - 4 r^2))
5 - 3 = 2:
r^3 (8 s^3 r^3 - 9 s^4 r^2 + 3 s^5 r^(4 - 3) - s^4 (2 r s - 5 s^2 - 4 r^2))
4 - 3 = 1:
r^3 (8 s^3 r^3 - 9 s^4 r^2 + 3 s^5 r - s^4 (2 r s - 5 s^2 - 4 r^2))
Factor s^3 out of 8 s^3 r^3 - 9 s^4 r^2 + 3 s^5 r - s^4 (2 r s - 5 s^2 - 4 r^2), resulting in s^3 (8 r^3 - 9 r^2 s^(4 - 3) + 3 r s^(5 - 3) - (2 r s - 5 s^2 - 4 r^2) s^(4 - 3)):
r^3 s^3 (8 r^3 - 9 r^2 s^(4 - 3) + 3 r s^(5 - 3) - s^(4 - 3) (2 r s - 5 s^2 - 4 r^2))
4 - 3 = 1:
r^3 s^3 (8 r^3 - 9 r^2 s + 3 r s^(5 - 3) - s^(4 - 3) (2 r s - 5 s^2 - 4 r^2))
5 - 3 = 2:
r^3 s^3 (8 r^3 - 9 r^2 s + 3 r s^2 - s^(4 - 3) (2 r s - 5 s^2 - 4 r^2))
4 - 3 = 1:
r^3 s^3 (8 r^3 - 9 r^2 s + 3 r s^2 - s (2 r s - 5 s^2 - 4 r^2
-s (-4 r^2 + 2 r s - 5 s^2) = 4 r^2 s - 2 r s^2 + 5 s^3:
r^3 s^3 (8 r^3 - 9 r^2 s + 3 r s^2 + 4 r^2 s - 2 r s^2 + 5 s^3)
Grouping like terms, 8 r^3 - 9 r^2 s + 3 r s^2 + 4 r^2 s - 2 r s^2 + 5 s^3 = 5 s^3 + (3 r s^2 - 2 r s^2) + (4 r^2 s - 9 r^2 s) + 8 r^3:
r^3 s^3 (5 s^3 + (3 r s^2 - 2 r s^2) + (4 r^2 s - 9 r^2 s) + 8 r^3)
3 (r s^2) - 2 (r s^2) = r s^2:
r^3 s^3 (5 s^3 + r s^2 + (4 r^2 s - 9 r^2 s) + 8 r^3)
4 (r^2 s) - 9 (r^2 s) = -5 (r^2 s):
Answer: r^3 s^3 (5 s^3 + r s^2 + -5 r^2 s + 8 r^3)