If `a^(2)-2a-3` is a factor of `a^(4)+pa^(3)+12a-9`, find the value of `p^(2)-2q-3`.

Explanation

We have`a^(2)-2a-3` is a factor of `a^(4)+pa^(3)+qa^(2)+12a-9`. But `" " a^(2)=2a-3=(a-3)(a+1)`
`:.` On putting a-3=0 i.e.., a=3, we get the remainder zero. ltbr. `:. " " (3)^(4)+p(3)^(3)+q(3)^(2)+12(3)-9=0`
`implies " " 81+27p+q+36-9=0`
`implies " " 27p+9q=-108`
`implies " " 3p+q=-12 " " ..........(1)`
Alos (a+1) is a factor of `a^(4)+pa^(3)+qa^(2)+12a-9`
`:.` On putting a+1=0 i.e.., a=-1, we get remainder=0
`:. (-1)^(4)+p(-1)^(3)+q(-1)^(2)+12^(-1)-9=0`
`implies " " 1-p+q-12-9=0`
`implies " "p-q=-20 " "`........(2)
Adding equations (1) and (2), we get
4p=-32
`:. " "p=-8`
Putting p=-8 in equation (1), we get
`3(-8)+q=-12`
`implies " " q=12`
`:. " " p^(2)-2q-3=(-8)^(2)-2(12)-3`
=64-24-3=37 `" "` Ans.