Question

Biết: a+b+c+d = 0; (a+c)³ = -(b+d)³

Chứng minh rằng: a³+b³+c³+d³ = 3(b+d)(ac-bd)

Answer 1

Giải:

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

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

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

Ta có:

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

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

\(\Leftrightarrow a^3+3a^2c+3ac^2+c^3=-b^3-3b^2d-3bd^2-d^3\)

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

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

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

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

Vậy ...