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Higher Order Equations

Let P(x) = x^...

Question

Let $P (x) = x^{2} + b x + c$ , where $b$ and $c$ are integers. If $P (x)$ is a facter of both $x^{4} + 6 x^{2} + 25$ and $3 x^{4} + 4 x^{2} + 28 x + 5$ , then value of $P (1)$ is -

A

4

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B

8

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C

10

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D

12

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Solution

The correct option is A $4$
Since, $P (x) = x^{2} + b x + c$ , where $b$ and $c$ are integers. If $P (x)$ is a facter of both $x^{4} + 6 x^{2} + 25$ and $3 x^{4} + 4 x^{2} + 28 x + 5$
Therefore, $P (x)$ will be factor of $(3 x^{4} + 4 x^{2} + 28 x + 5) - 3 (x^{4} + 6 x^{2} + 25) = - 14 (x^{2} - 2 x + 5)$
So $P (x) = x^{2} - 2 x + 5$
$P (1) = 1 - 2 + 5 = 4$

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