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

Let `p(x) = x^3 + 10x^2 + ax + b`
` p(x)` i.e. `x^3 + 10x^2 + ax + 6` is exactly divisible by `(x – 1) `and `(x + 2)`
Therefore, `p(1) `and `p(-2)` must equal to zero.
`p(1) = (1)^3 + 10(1)^2 + a x 1 + b = 0 `
`1 + 10 + a + b = 0`
` a + b = – 11 …(i)`
Also, `p(-2) = 0`
` (-2)^3 + 10(-2)^2 + a x (- 2) + b = 0`
` -8 + 40 – 2a + b = 0 `
`– 2a + b = – 32 …(ii)`
Solving (i) and (ii), we get
` a = 7` and `b = -18`