Cho 1 a + 1 b + 1 c = 0 . Tính giá trị biểu thức P = a b c 2 + b c a 2 + a c b 2

Đề bài: Cho $\frac{1}{a} +\frac{1}{b} +\frac{1}{c} = 0$ . Tính giá trị biểu thức 

Trả lời

Hướng dẫn giải:

$$\begin{gathered} P=\frac{ab}{c^2}+\frac{bc}{a^2}+\frac{ac}{b^2} \\ \Rightarrow P=\frac{abc}{c^3}+\frac{abc}{a^3}+\frac{abc}{b^3} \\ \Rightarrow P=abc\left(\frac{1}{c^3}+\frac{1}{a^3}+\frac{1}{b^3}\right) \end{gathered}$$

Vì $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$

$$\begin{gathered} \Rightarrow \frac{1}{a}+\frac{1}{b}=-\frac{1}{c} \\ \Rightarrow {\left(\frac{\text{1}}{\text{a}}\text{+}\frac{\text{1}}{\text{b}}\right)}^{\text{3}}\text{=}{\left(-\frac{\text{1}}{\text{c}}\right)}^{\text{3}} \\ \Rightarrow \frac{\text{1}}{{\text{a}}^{\text{3}}}\text{+}\frac{\text{1}}{{\text{b}}^{\text{3}}}\text{+}\frac{\text{3}}{\text{ab}}\left(\frac{\text{1}}{\text{a}}\text{+}\frac{\text{1}}{\text{b}}\right)\text{=}-\frac{\text{1}}{{\text{c}}^{\text{3}}} \\ \Rightarrow \frac{\text{1}}{{\text{a}}^{\text{3}}}\text{+}\frac{\text{1}}{{\text{b}}^{\text{3}}}\text{+}\frac{\text{1}}{{\text{c}}^{\text{3}}}\text{+}\frac{\text{3}}{\text{ab}}\left(-\frac{\text{1}}{\text{c}}\right)\text{=0} \\ \Rightarrow \frac{\text{1}}{{\text{a}}^{\text{3}}}\text{+}\frac{\text{1}}{{\text{b}}^{\text{3}}}\text{+}\frac{\text{1}}{{\text{c}}^{\text{3}}}\text{=}\frac{\text{3}}{\text{abc}}\left(\text{1}\right) \end{gathered}$$

Thay (1) vào P ta được

$\text{P = abc.}\frac{\text{3}}{\text{abc}}$

⇒ P = 3.