Polynomials/Polynomials

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Problem: Find rational roots of a polynomial: \[ P(x)=-9 x^{4}+6 x^{3}-23 x^{2}+4 x+4 . \]

1.10.1 Polynomials

1.55 $

Problem: Find rational roots of a polynomial: \[ P(x)=2 x^{4}-5 x^{3}+4 x^{2}+3 x+9 . \]

1.10.2 Polynomials

1.55 $

Problem: Find GCD of polynomials \( f(x), g(x) \in \mathrm{F}_{2}[x] \), where \[ \begin{array}{l} f(x)=x^{7}+x^{5}+x^{4}+x+1, \\ g(x)=x^{8}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+1 . \end{array} \]

1.10.3 Polynomials

2.58 $

Problem: Solve the system of equations in finite fields \( \mathbb{Z}_{7} \) and \( \mathbb{Z}_{11} \) : \[ \left\{\begin{array}{l} 5 x+3 y=1 \\ 3 x-2 y=2 \end{array}\right. \]

1.10.4 Polynomials

2.06 $

Problem: Solve an algebraic equation in finite fields \( \mathbb{Z}_{5} \) and \( \mathbb{Z}_{7} \). \[ x^{3}+x^{2}+3 x+3=0 \text {. } \]

1.10.5 Polynomials

3.1 $

Problem: Decompose the polynomial \( g(x)=1+g_{1} x+ \) \( +g_{2} x^{2}+g_{3} x^{3}+g_{4} x^{4}+x^{5} \in \mathbb{Z}_{2}[x] \) into a product of irreducible factors. \[ g_{1} g_{2} g_{3} g_{4}=1100 \text {. } \]

1.10.6 Polynomials

2.06 $