[Community Question] Algebra-Precalculus: Roots of polynomials and their formulae relating to coefficients

One of our user asked:

Write down the cubic equation given that $\alpha + \beta + \gamma = 4$, $\alpha^2 + \beta^2 + \gamma^2 = 66$, and $\alpha^3 + \beta^3 + \gamma^3 = 280$

Ok so, the sum of roots is given and I'm able to use the sum of the roots and the sum of the roots squared to get the sum of the combination of roots, but I'm unable to get the product of roots, because I can't seem to manipulate the sum of the cubes of roots to resemble the sum of roots and sum of squares of roots.

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[Community Question] Linear-algebra: non-negative matrix satisfying two conditions

One of our user asked: A real matrix $B$ is called non-negative if every entry is non-negative. We will denote this by $B\ge 0$ . I want to find a non-negative matrix $B$ satisfying the following two conditions: (1) $(I-B)^{-1}$ exists but not non-negative. Here $I$ is the identity matrix. (2) There is a non-zero and non-negative vector $\vec{d}$ such that $(I-B)^{-1}\vec{d}\ge 0$ . I tried all the $2\times 2$ matrices, but it did not work. I conjecture that such a $B$ does not exist, but don't know how to prove it.

[Community Question] Geometry: The limit about the line connecting the intersection of a circle and the $y$-axis and the intersection of the shrinking circle and a fixed circle

One of our user asked: There is a fixed circle $C_1$ with equation $(x - 1)^2 + y^2 = 1$ and a shrinking circle $C_2$ with radius $r$ and center the origin. $P$ is the point $(0, r)$ , $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ and the $x$ -axis. What happens to $R$ as $C_2$ shrinks, that is, as $r \to 0^+$ ? (The figure is made with GeoGebra ) In order to solve this problem, I made a script using GeoGebra in which the circle $C_2$ is a dynamic one whose radius $r$ can be adjusted with a slider. As I set $r \to 0^+$ , the figure seems to suggest that $R \to (4,0)$ . In particular, this is the state with $r = 0.001$ , in which $R$ is reported to be $(3.9999997523053,0)$ : However, I would like to find out a way to prove (or disprove, though unlikely) my guess that $$\lim_{r \to 0} R = (4,0).$$ But I have little idea. Any help would be appreciated.

Order of elements of the Prüfer groups $\mathbb{Z}(p^{\infty})$

Let $\mathbb{Z}(p^{\infty})$ be defined by $\mathbb{Z}(p^{\infty}) = \{ \overline{a/b} \in \mathbb{Q}/ \mathbb{Z} / a,b \in \mathbb{Z}, b=p^i$ $ with$ $ i \in \mathbb{N} \}$ , I wish show that any element in $\mathbb{Z}(p^{\infty})$ has order $p^n$ with $n \in \mathbb{N}$ . i try several ways but I have not been successful, some help ?? thank you

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