InterviewSolution
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InterviewSolution
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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)`.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 |
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Answer» Correct Answer - 3 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)` |
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