InterviewSolution
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InterviewSolution
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Consider an unknow polynomial which divided by `(x - 3)` and `(x-4)` leaves remainder 2 and 1, respectively. Let R(x) be the remainder when this polynomial is divided by `(x-3)(x-4)`. If equations `R(x) = x^(2)+ ax +1` has two distint real roots, then exhaustive values of a are.A. `(-2,2)`B. `(-oo,-2) uu(2,oo)`C. `(-2,oo)`D. all real numbers |
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Answer» Correct Answer - 4 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` `5-x = x^(2) + ax + 1 ` `rArr x^(2) + (a + 1) x - 4 =0` Given that roots and reall distinct. Therefore, `D lt 0 rArr (a+1)^(2) + 16 lt 0` Which is true for all real a. |
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