\(B=\dfrac{a^2}{\left(a-b\right)\left(a-c\right)}+\dfrac{b^2}{\left(b-a\right)\left(b-c\right)}+\dfrac{c^2}{\left(c-a\right)\left(c-b\right)}\)
\(\Rightarrow B=\dfrac{a^2}{\left(a-b\right)\left(a-c\right)}-\dfrac{b^2}{\left(a-b\right)\left(b-c\right)}+\dfrac{c^2}{\left(a-c\right)\left(b-c\right)}\)
\(\Rightarrow B=\dfrac{a^2\left(b-c\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}-\dfrac{b^2\left(a-c\right)}{\left(a-b\right)\left(b-c\right)\left(a-c\right)}+\dfrac{c^2\left(a-b\right)}{\left(a-c\right)\left(b-c\right)\left(a-b\right)}\)
\(\Rightarrow B=\dfrac{a^2b-a^2c}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}-\dfrac{ab^2-b^2c}{\left(a-b\right)\left(b-c\right)\left(a-c\right)}+\dfrac{ac^2-bc^2}{\left(a-c\right)\left(b-c\right)\left(a-b\right)}\)
\(\Rightarrow B=\dfrac{a^2b-a^2c-ab^2+b^2c+ac^2-bc^2}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)
\(\Rightarrow B=\dfrac{\left(a^2b-ab^2\right)-\left(a^2c-abc\right)-\left(abc+b^2c\right)+\left(ac^2-bc^2\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)
\(\Rightarrow B=\dfrac{ab\left(a-b\right)-ac\left(a-b\right)-bc\left(a-b\right)+c^2\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)
\(\Rightarrow B=\dfrac{\left(ab-ac-bc+c^2\right)\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)
\(\Rightarrow B=\dfrac{\left[a\left(b-c\right)-c\left(b-c\right)\right]\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)
\(\Rightarrow B=\dfrac{\left(a-c\right)\left(b-c\right)\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)
\(\Rightarrow B=1\)
