Find the values of a and b so that `2x^(3)+ax^(2)+bx-14` has (x-1) and (x+2) are its factors.

Explanation

Let `p(x)=2x^(3)+ax^(2)+bx-14`
If (x-1) is a factor of (1), then f(1)=0
`therefore` On putting x=1 in (1), we get
`f(1)=2(1)^(3)+a(1)^(2)+b(1)-14`
`implies 0=2+a+b-14 implies a+b=12`
If (x+2) is a factor of (1), then f(-2)=0
`therefore` On putting x=-2 in (1), we get
`f(-2)=2(-2)^(3)+a(-2)^(2)+b(-2)-14`
`implies 0 =-16+4a-2b-14`
`implies Aa-2b=30 implies 2a-b=15`
Adding (2) and (3), we get
`3a=27 " " implies a=9`
Put this value of a in (2), we get
`9+b=12 " " implies b=3`
`therefore a=9, b=3`.