Question

cho 1/a+1/b+1/c=0

và M=b^2c^2/a+c^2a^2/b+a^2b^2/c

c/m M=3abc

 

 

 

Answer 1

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\)

<=>  \(\frac{ab+bc+ca}{abc}=0\)

<=>  \(ab+bc+ca=0\)

=>  \(ab+bc=-ca\)

<=>  \(\left(ab+bc\right)^3=-ca^3\)

Ta co:   \(a^3b^3+b^3c^3+c^3a^3=a^3b^3+b^3c^3-\left(ab+bc\right)^3=a^3b^3+b^3c^3-ab^3-bc^3-3ab.bc\left(ab+bc\right)\)

\(=-3ab.bc\left(ab+bc\right)=-3ab.bc.\left(-ca\right)=3a^2b^2c^2\)

\(M=\frac{b^2c^2}{a}+\frac{c^2a^2}{b}+\frac{a^2b^2}{c}=\frac{b^3c^3+c^3a^3+a^3b^3}{abc}=\frac{3a^2b^2c^2}{abc}=3abc\)