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Question


What number should be subtracted from the expression P=x3+4x27x+12 for it to be perfectly divisible by x + 3?


Options:
A .   42
B .   39
C .   13
D .   None of these
Answer: Option A
:
A

Method 1 :-
According to remainder theorem, the remainder for f(x)x+a is f(-a).
Let the number to be subtracted be 'k'.
Since, x+3 divides x3+4x27x+12k perfectly the remainder when x is substituted by -3 is 0f(3)=0.
(3)3+4(3)27(3)+12k=0
27+36+21+12=k
k=42
Method 2 :-
If we want x3+4x27x+12 to be perfectly divisible by x+3,
x3+4x27x+12 should be of the form of (x+3)×g(x).
(x3+4x27x+12) 
=(x+3)(x2+x10)+12+30
=(x+3)(x2+x10)+42
So, the number that needs to be subtracted = 42.
Method 3 :- Shortcut!
Since this is a variable-number question, assume x=1; then the expression P=10.
Now we need to check which answer option satisfies the following condition.
10(answer option)4(Remainder)=0
10424(Remainder)=324(Remainder)=0



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