Let a, b , and c be such that 1 (1-x)(1-2 x)(1-3 x) = a 1-x + b 1-2 x + c 1-3 x then a 1 + b 3 + c 5 is equal to

Step-by-step solution

We have, 1=a(1-2 x)(1-3 x)+b(1-x)(1-3 x)+c(1-x)(1-2 x) Put x= 1 2 1=0+b (1- 1 2 ) (1- 3 2 )+0 1=b ( 1 2 ) (- 1 2 ) b=-4 Put x=1 , 1=a(1-2)(1-3)+0+0 1=a(-1)(-2) a= 1 2 Put x= 1 3 , 1=0+0+c (1- 1 3 ) (1- 2 3 ) 1=c ( 2 3 ) ( 1 3 ) c= 9 2 Now, a 1 + b 3 + c 5 = 1 2 + (-4) 3 + 9 5 2 = 1 2 - 4 3 + 9 10 = 15-40+27 30 = 2 30 = 1 15