Question

Cho a+ b +c + d = 0. Chứng minh rằng a³ + b³ +c³ + d³ = 3 ( c +d ) ( ab - cd )

giúp với mik cần gấp trong hôm nay

Answer 1

Giải:

Ta có:

\(a+b+c+d=0\)

\(\Leftrightarrow a+b=-c-d\)

\(\Leftrightarrow a+b=-\left(c+d\right)\)

Từ đó, suy ra:

\(\left(a+b\right)^3=-\left(c+d\right)^3\)

\(\Leftrightarrow a^3+3a^2b+3ab^2+b^3=-\left(c^3+3c^2d+3cd^2+d^3\right)\)

\(\Leftrightarrow a^3+3a^2b+3ab^2+b^3=-c^3-3c^2d-3cd^2-d^3\)

\(\Leftrightarrow a^3+3ab\left(a+b\right)+b^3=-c^3-3cd\left(c+d\right)-d^3\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=-3cd\left(c+d\right)-3ab\left(a+b\right)\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=-3cd\left(c+d\right)+3ab\left(c+d\right)\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=3\left(c+d\right)\left(ab-cd\right)\)

Vậy ...