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Question

Let f(x)=x5+ax4+bx3+cx2+dx420 where a,b,c,d are real parameters, be a polynomial. If all zeros of the polynomial f(x) are integers larger than 1, and f(4) is equal to k, then k is divisible by

A
2
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B
3
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C
5
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D
6
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Solution

The correct option is D 6
f(x)=0 has 5 integral (not necessarily distinct) roots d1,d2,...,d5
Then f(x)=(xd1)(xd2)(xd3)(xd4)(xd5)
Product of roots, d1d2d3d4d5=420
and 420=22357

All of the roots are integers larger than 1, so they must be 2,2,3,5 and 7.
So f(x)=(x2)2(x3)(x5)(x7)
Putting x=4 gives k=12.

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