The correct option is C 3x
f(x)=x5+ax3+bx
When f(x) is divided by x+1, the remainder is −3, so
f(−1)=−3
⇒−1−a−b=−3⇒a+b=2 ⋯(1)
Assuming
f(x)=p(x)×(x2−1)+qx+r
Putting x=−1, we get
⇒−3=−q+r ⋯(2)
Putting x=1, we get
⇒f(1)=q+r⇒1+a+b=q+r
Using equation (1), we get
3=q+r ⋯(3)
Solving equations (2) and (3), we get
q=3, r=0
Therefore, f(x)=p(x)×(x2−1)+3x
Hence, the remainder is 3x.