$x^4 -ax^3 +2x^2 -bx +1$ has real root $\implies$ $a^2+b^2 \ge 8$
5410
Prove that $a^2 + b^2 \geq 8$ if $ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $ has at least one real root.
85525
Prove that $x^3+x+1$ splits in $Z_2[x]/(x^3+x+1)$
15730
Determine the minimum of $a^2 + b^2$ if $a,b\in\mathbb{R}$ are such that $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has at least one real solution
92692
Prove $3(x^2y+y^2z+z^2x)(xy^2+yz^2+zx^2)\ge xyz(x+y+z)^3$
21972
How to show that $\sqrt{2}+\sqrt{3}$ is algebraic?
80304
Roots of $\,ax^2+bx+c\,$ are irrational if $a,b,c$ are odd integers
19909
if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$
106374
Show that $2x^6+12x^5+30x^4+60x^3+80x^2+30x+45=0$ has no real roots
38662
If $a,b$ are the roots of the equation $2 x^2 -3 x +1 = 0$, find an equation whose roots are $a/(2b +3)$, $b/(2a +3)$
111295
Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$
155165
Prove that $\mathbb F_8=\mathbb F_2[X]/(X^3+X+1)$
55119
Proof that the coefficients of a polynomial are real
141184
Proof that a degree 4 polynomial has at least two roots
74222
If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$
49548
If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
97750
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $a^2 + b^2 + c^2 \ge a + b + c$.
55197
Roots of $ x^3-3x+1$