If a polynomial f (x) is divided by ` (x - 3) and (x - 4)` it leaves remainders as 7 and 12 respectively, then find the remainder when f (x) is divided by `(x-3)(x-4)`.

Options

A. `[-2,2]`
B. `(-oo,-2- sqrt(3)] uu [-2+ sqrt(3),oo)`
C. `(-oo,-2-3sqrt(3)] uu [-2+sqrt(3),oo)`
D. none of these

Answer

3

Explanation

Let unknow polynomial be P(x). Let Q(x) and R(x) be the quatient and remainder, respectively, when it is divided by the `(x-3) (x-4)`. Then
`P(x) = (x-3)(x-4) Q(x) + R(x)`
Then we have
`R(x) = ax + b`
` rArr P(x) = (x-3) (x-4)Q(x) + ax+ b`
Given that `P(3) =2 ` and `P(4)=1` . Hence,
`3a + b = 2 and 4a + b = 1`
`rArr a = -1 and b= 5`
`rArr R(x) = 5 -x`
`f(x)= y = (-x + 5)/(x^(2) - 3x + 2)`
`rArr yx^(2) + (1-3y) x + 2y - 5 = 0`
Now, x is real, then
`D ge 0`
`rArr (1- 3y) ^(2) - 4y(2y - 5) ge 0`
` or y^(2) + 14y + 1 ge 0`
` y in (-oo, (-14- sqrt(192))/(2)]uu[(-14+sqrt(192))/(2),oo)`
`(-oo, -7-4sqrt(3)] uu [ - 7 + 4sqrt(3),oo)`