Question

$I^3(m-n{)}^3+m^3(n-1{)}^3+n^3(I-m{)}^3$ equals to ______.

• A. $3Imn(m-n)(n-1)(I-m)$
• B. 0
• C. none of these
• D. $\left(IImn{)}^3\right(I-m\left)\right(m-n\left)\right(n-1)$
• E. $(Im+mn+nl)\left(3I^2{M}^2{N}^2\right)$

Answer 1

Answer: (C)

$I^3(m-n{)}^3+m^3(n-1{)}^3+n^3(I-m{)}^3=I^3[(m{)}^3-(n{)}^3-I\left(mn\right)(m-n)]+m^3(n{)}^3-(1{)}^3-3\left(x\right)(x-1)+n^3(I{)}^3-(m{)}^3\left(mI\right)(I-m)=I^3[m^3-n^3-3mn(m-n)]+m^3[n^3-1-3n(n-1)]+n^3[I^3-m^3-3mI(I-m)]=m^3{I}^3-n^3{I}^3-3m^{2}n{I}^3+3m{n}^2{I}^2+m^3{n}^3-m^3-3n^2{m}^3+3n{m}^3+n^3{I}^3-n{m}^3-3m{I}^2{n}^3+3m^2{n}^3$