If the polynomials `az^3+4z^2+3z-4` and `z^3-4z+a` leave the same remainder when divided by `z-3` , find the value of a.

Options

A. 1
B. -2
C. -1
D. 2

Answer

C

Explanation

Let `p_(1)(z) =az^(3) +4z^(2) +3z -4 and P_(2) (z)=z^(3)-4z+a`
when we divide `p_(1)(z) "by" z-3` ,then we get the remainder `p_(1)(3)`.
Now `P_(1) (3)=a(3)^(3) +4(3)^(2)+3(3)-4`
`=27a+36+9-4=27a+41`
when we divide `p_(2)(z)byz-3` , then we get the remainder `p_(2)(3).`
Now` p_(2) (3)=(3)^(3) -4(3)+a`
`=27-12+a=15+a`
According to the question , both the remainders are same .
i.e., `p_(1)(3)=P_(2)(3)`
`therefore 27a+41=15+a`
`implies 27a-a =15-41`
`implies 26a=26`
`implies a=-1`