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Q.
If $p(x) =x^{2}+bx+c$, where $b, c \in\,I$ Ifp (x) is a factor of both $x^{4}+6x^{2}+25$ and $3x^{4}+4x^{2}+28x+5$, then $P(2)$ equals
Complex Numbers and Quadratic Equations
Solution:
$p(x)$ is a factor of $x^{4}+6x^{2}+25=g (x)$ (say)
and $3x^{4}+4x^{2}+28x+5 = h (x)$ (say)
then p(x) must be the factor of $h (x) -k \,g (x) \forall\,x \, \in\,R$
$\therefore =(3x^{4}+4x^{2}+28x+5)-3(x^{4}+6x^{2}+25)$
$=-14(x^{2}-2x+5)$
$\therefore p(x) = x^{2}-2x+5$
$\therefore p(2) =4-4+5=5$