Question

If f(x)f(x)  is divided byx−3x−3  then the remainder is 55 , and when it is divided by (x−4)(x−4)  the remainder is '0''0' . Then the remainder when it is divided by (x−3)(x−4)(x−3)(x−4)  is

A.   

−5x+20−5x+20

B.   

5x−205x−20

C.   

5x+205x+20

D.   

−(5x+20)−(5x+20)


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Answer 1

Answer:

option D : −(5x+20)

Step-by-step explanation:

f(x) when divided by x - 3 and x - 4 leaves remainder 5 and 0 respectively.

From polynomial-remainder theorem,

P(3) = 5 and P (4) = 0 If the polynomial is divided by (x – 3)(x - 4)

then remainder must be of the form ax + b (degree of remainder is less than that of divisor)

f(x) = Q(x)(x - 3)(x - 4) + (ax + b)

where Q(x) is some polynomial. Substituting for x = 3 and x = 4

P(3) = 5 = 3a + b

P(4) = 0 = 4a + b

Solving for a and b, we get

a = -5 and b = -20

Remainder = -5x - 20 = −(5x+20)