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Polynomial Division and Remainder Theorem - Class IX
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If $P(x)$ and $Q(x)$ are two polynomial such that $f(x)=P(x^3)+Q(x^3)$ is divisible by $x^2+x+1$, then?
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A
Both $P(x)$ and $Q(x)$ are divisible by $(x-1)$
💡 Explanation:
For f(x) to be divisible by x^2+x+1, it must vanish at the roots of x^2+x+1=0, which are omega and omega^2. Substituting x=omega into P(x^3)+Q(x^3) gives P(1)+Q(1)=0, implying P(1)=-Q(1). This condition is satisfied if both P(x) and Q(x) contain the factor (x-1).