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Question

Let P(x)=x2+bx+c, where b and c are integers. If P(x) is a facter of both x4+6x2+25 and 3x4+4x2+28x+5, then value of P(1) is -

A
12
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B
4
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C
10
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D
8
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Solution

The correct option is B 4
P(x)=x2+bx+c is a factor ofx4+6x2+25.

x4+6x2+25=x4+10x24x2+25=(x4+10x2+25)4x2=(x2+5)2(2x)2

x4+6x2+25=(x2+2x+5)(x22x+5)

One of them is a factor of 3x4+4x2+28x+5.

3x4+4x2+28x+5=3x46x3+15x2+6x311x2+28x+5

3x2(x22x+5)+6x312x2+30x+x22x+5=3x2(x22x+5)+6x(x22x+5)+x22x+5

3x4+4x2+28x+5=(x22x+5)(3x2+6x+1)

So, P(x)=x22x+5
P(1)=12+5=4

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