Skip to main content

[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.


Labels:

Comments

Post a Comment

Popular posts from this blog

[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.
Post a Comment
Read more

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
Post a Comment
Read more