# VMath

### Основное

#### Действия

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Satellite page for DISCRIMINANT

# Problems

1. Find the parameter value ${\color{Red} \alpha } \in [-5,5]$ for which $$\max_{x\in]-\infty,+\infty[} (-x^6+{\color{Red} \alpha } \,x^5+3\,x^3+5\,x^2-2\,x-1)$$ is minimal. Evaluate this minimum.

2. Establish the validity or invalidity of the following equality $$\mathcal D_x(\mathcal D_y (f(x,y))=\mathcal D_y(\mathcal D_x (f(x,y)) \, .$$ Here $\mathcal D_x$ and $\mathcal D_y$ denotes the discriminants of the polynomials treated with respect to the indexed variables. Compute $$\gcd(\mathcal D_x(\mathcal D_y (f(x,y)),\mathcal D_y(\mathcal D_x (f(x,y))) \quad \mbox{ for } \quad f(x,y)= -x^4-2\,y^4+4\,x^2-{\color{Red} \alpha } \,y^2+3\,xy+4\,y \ .$$

3. Prove that the discriminan of the polynomial $f(x)=x^4+p\,x^2+q\,x+r$ coincides with the discriminant of its Ferrari's resolvent $$\mathcal D_x(f(x))= \mathcal D_t( t^3-p\,t^2-4\,r\,t+(4\,pr-q^2)) \ .$$

4. Prove that if the polynomial of the degree $4$ has its discriminant positve and two of its roots real, then all its roots are real.

5. Prove that $\mathcal D (f(x)f^{\prime \prime}(x)- [f^{\prime}(x)]^2)$, treated as a polynomial with respect to the coefficients of $f(x)$, is divisible by $\mathcal D (f(x))$.

detse/discrime/problemse.txt · Последние изменения: 2020/11/04 23:59 — au