Given, (x−1)(x−2)(x−3)x4 =x+k+x−1A+x−2B+x−3C ⇒x4=x(x−1)(x−2)(x−3)+k(x−1) (x−2)(x−3)+A(x−2)(x−3)+B(x−3) (x−1)+C(x−1)(x−2)
on putting x=1, we get A=21
on putting x=2, we get B=−116=−16
on putting x=3, we get C=281
on putting x=0, we get 0=k(−1)(−2)(−3)+21(−2)(−3)−16 (−3)(−1)+281(−1)(−2) ⇒6k=3−48+81=36 ⇒k=6 ∴k+A−B+C=6+21+16+281 =6+16+41=63