If f(x) and g(x) are two polynomials such tha | vedclass

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Question

$P(x)=f\left(x^{3}\right)+\operatorname{xg}\left(x^{3}\right)$

$P (1)=f(1)+ g (1) .....(1)$

Now $P ( x )$ is divisible by $x ^{2}+ x +1$

$\Ri

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$P(x)=f\left(x^{3}\right)+\operatorname{xg}\left(x^{3}\right)$

$P (1)=f(1)+ g (1) .....(1)$

Now $P ( x )$ is divisible by $x ^{2}+ x +1$

$\Rightarrow P ( x )= Q ( x )\left( x ^{2}+ x +1\right)$

$P ( w )=0= P \left( w ^{2}\right)$ where $w , w ^{2}$ are non-real cube roots of units

$P ( x )=f\left( x ^{3}\right)+ xg \left( x ^{3}\right)$

$P ( w )=f\left( w ^{3}\right)+ wg \left( w ^{3}\right)=0$

$f(1)+\operatorname{wg}(1)=2 .....(2)$

$P \left( w ^{2}\right)=f\left( w ^{6}\right)+ w ^{2} g \left( w ^{6}\right)=0$

$f(1)+w^{2} g(1)=0 .....(3)$

$(2)+(3)$

$\Rightarrow 2 f(1)+\left(w+w^{2}\right) g(1)=0$

$2 f(1)= g (1) .....(4)$

$(2)-(3)$

$\Rightarrow\left( w - w ^{2}\right) g (1)=0$

$g(1)=0=f(1) \quad$ from (4)

from $1) \,P (1)=f(1)+ g (1)=0$

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P ( x )=f\left( x ^{3}\right)+ xg \left( x ^{3}\right)$ is divisible by $x^{2}+x+1,$ then $P(1)$ is equal to ....... .

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