1/[(a-x)(b-x)(c-x)

1/[(a-x)(b-x)(c-x)]≡ A/(a-x) +B/(b-x)+C/(c-x)
=>
1≡ A(b-x)(c-x) +B(a-x)(c-x)+C(a-x)(b-x)
x=a =>A= 1/(b-a)(c-a)
x=b =>B= 1/(a-b)(c-b)
x=c =>C= 1/(a-c)(b-c)

ie
1/[(a-x)(b-x)(c-x)]
= [1/(b-a)(c-a)][1/(a-x)] +[1/(a-b)(c-b)][1/(b-x)]+[1/(a-c)(b-c)][1/(c-x)]