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\(\frac{a^3}{\left(a-b\right)\left(a-c\right)}+\frac{b^3}{\left(b-c\right)\left(b-a\right)}+\frac{c^3}{\left(c-a\right)\left(c-b\right)}\)
\(=\frac{a^3\left(b-c\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}+\frac{b^3\left(c-a\right)}{\left(b-c\right)\left(a-b\right)\left(a-c\right)}+\frac{c^3\left(a-b\right)}{\left(a-c\right)\left(b-c\right)\left(a-b\right)}\)
\(=\frac{a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)
\(=\frac{\left(a-b\right)\left(a-c\right)\left(b-c\right)\left(a+b+c\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}=a+b+c\)
\(=\frac{a^3\left(b-c\right)-b^3\left(a-c\right)+c^3\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}=\frac{a^3b-ab^3-a^3c+b^3c+c^3\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)
\(=\frac{ab\left(a^2-b^2\right)-c\left(a^3-b^3\right)+c^3\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}=\frac{ab\left(a+b\right)-c\left(a^2+b^2+ab\right)+c^3}{\left(a-c\right)\left(b-c\right)}\)
\(=\frac{a^2b+ab^2-a^2c-b^2c-abc+c^3}{\left(a-c\right)\left(b-c\right)}=\frac{a^2\left(b-c\right)+ab\left(b-c\right)-c\left(b^2-c^2\right)}{\left(a-c\right)\left(b-c\right)}\)
\(=\frac{a^2+ab-c\left(b+c\right)}{a-c}=\frac{a^2+ab-bc-c^2}{a-c}=\frac{b\left(a-c\right)+\left(a^2-c^2\right)}{a-c}=a+b+c\)
\(VT=\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}\)
\(=\frac{\left(b-a\right)-\left(c-a\right)}{\left(b-a\right)\left(c-a\right)}+\frac{\left(c-b\right)-\left(a-b\right)}{\left(c-b\right)\left(a-b\right)}+\frac{\left(a-c\right)-\left(b-c\right)}{\left(a-c\right)\left(b-c\right)}\)
\(=\frac{1}{c-a}-\frac{1}{b-a}+\frac{1}{a-b}-\frac{1}{c-b}+\frac{1}{b-c}-\frac{1}{a-c}\)
\(=\frac{1}{c-a}+\frac{1}{a-b}+\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{b-c}+\frac{1}{c-a}\)
\(=2\left(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}\right)=VP\left(đpcm\right)\)
A=a^3(b-c)+b^3(c-a)+c^3(a-b)/(a-b)(a-c)(b-c)
phân tích mẫu thức thành nhân tữ được (a-b)(b-c)(a-c)(a+b+c)
=>A=a+b+c=2016
bạn đổi dấu ở mẫu ý rồi quy đồng lên
rồi cộng vào phân tích là ra đấy
thông cảm máy mk ko thể trình bày rõ đc
chỉ nói ý đc thui:))