If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x³) + x g(x³) is divisible by x² + x + 1, then P(1) is equal to ___________.

Solution

Maths sir · 4 steps

Put x = ω and x = ω²: since ω³ = 1, both f and g are evaluated at 1.

1. Roots of the divisor. The roots of x² + x + 1 = 0 are ω and ω², the complex cube roots of unity. Since P(x) is divisible by x² + x + 1, we get P(ω) = 0 and P(ω²) = 0.
2. Use omega cubed. Because ω³ = 1 and (ω²)³ = 1, every x³ inside f and g becomes 1. The two conditions are two linear equations in f(1) and g(1).

   f(1) + ω g(1) = 0, f(1) + ω² g(1) = 0

3. Solve the pair. Subtract the second equation from the first: (ω - ω²) g(1) = 0. Since ω ≠ ω², this forces g(1) = 0, and then the first equation gives f(1) = 0.
4. So. P(1) = f(1) + 1 · g(1) = 0 + 0 = 0.

The idea to keep. When x² + x + 1 divides a polynomial, substitute ω and ω². Any x³ inside the polynomial turns into 1, and you get simple equations to solve.

Answer from the NTA final answer key. Solution by PrepWiser.