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Question

Let P(x)=x2+bx+c, where b and c are integers. If P(x) is a factor of both x4+6x2+25 and 3x4+4x2+28x+5, then

A
P(1)=4
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B
P(1)=6
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C
P(x)=0 has roots of opposite sign
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D
P(x)=0 has imaginary roots
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Solution

The correct option is D P(x)=0 has imaginary roots
Since P(x) divides both the expressions, hence P(x) also divides (3x4+4x2+28x+5)3(x4+6x2+25)
=14x2+28x70
=14(x22x+5) which is a quadratic.
Hence, P(x)=x22x+5
P(1)=4

D=420=16<0
P(x)=0 has imaginary roots.

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