Question:medium
If \( f(x) \) and \( g(x) \) are two polynomials such that \( \phi(x) = f(x^3) + xg(x^3) \) is divisible by \( x^2 + x + 1 \), then:
If \( f(x) \) and \( g(x) \) are two polynomials such that \( \phi(x) = f(x^3) + xg(x^3) \) is divisible by \( x^2 + x + 1 \), then:
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Use the property that if a polynomial \( P(x) \) is divisible by \( (x - a) \), then \( P(a) = 0 \). The roots of \( x^2 + x + 1 = 0 \) are crucial here.
Updated On: Nov 28, 2025
- \( \phi(x) \) is divisible by \( (x - 1) \)
- none of \( f(x) \) and \( g(x) \) is divisible by \( (x - 1) \)
- \( g(x) \) is divisible by \( (x - 1) \) but \( f(x) \) is not divisible by \( (x - 1) \)
- \( f(x) \) is divisible by \( (x - 1) \) but \( g(x) \) is not divisible by \( (x - 1) \)
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