Find the values of `a` and `b` so that the polynomial `x^3+10 x^2+a x+b` is exactly divisible by `x-1` as well as `x-2.`

Explanation

`f(x)=x^3+10x^2+ax+b` is exactly divisible by `( x -1)` and `(x -2)`
Now,
Put `x=1`
`(1)^3+10(1)^2+a(1)+b=0`
`1+10+a+b=0`
`a+b=-11` --------(1)
Again put `x=2`
`(2)^3+10(2)^2+a(2)+b=0`
`8+40+2a+b=0`
`2a+b=-48` ----------(2)
From equation (1) and equation (2):
`(2a+b)-(a+b)=(-48)-(-11)`
`2a+b-a-b=-37`
`a=-37`
`a+b=11`
`-37+b=(-11)`
`b=37-11`
`b=26`
`therefore` The value of `a=-37` and `b=26`