5
votes
7answers
310 views

If $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$.

If $x>0$ and $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$. Is there some other way to do find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ ...
5
votes
1answer
82 views

Taking a power of a polynomial to make it symmetric

Suppose I have a non-symmetric multi-variable polynomial in $n$ variables $P(x_1,x_2,...,x_n)$. For example $P$ might be $x_1^2+x_2$ or $x_1-x_2$ Under what conditions will some power $m$ of $P$ ...
12
votes
11answers
1k views

How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations

I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong? $$(a-b)^3 + (b-c)^3 + (c-a)^3 ...
0
votes
1answer
51 views

How to prove this symmetric polynomial equations?

I got a problem from a friend, which is to prove that $\Sigma _{i=1}^{n}% \frac{x_{i}^{m}}{\Pi _{j\neq i}(x_{i}-x_{j})}=0$ for m < n-1. I tried to multiply the left of equation with $\Pi _{1\leq ...
1
vote
2answers
148 views

Three inequalities with sums of fractions over two positive integers

In a proof, I arrive at three inequalities for all $p,q \geqslant 0$: \begin{align} \frac{p+1}{q+1} + \frac{q+1}{p+1} &\geqslant 1 + \frac{p}{2q+1} + \frac{q}{2p+1} + \frac{1}{p+q+1};\cr ...
11
votes
1answer
2k views

Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...
4
votes
2answers
355 views

Need help solving a particular system of non-linear equations analytically

How would one go about analytically solving a system of non-linear equations of the form: $a + b + c = 4$ $a^2 + b^2 + c^2 = 6$ $a^3 + b^3 + c^3 = 10$ Thanks!
6
votes
3answers
662 views

Three-variable system of simultaneous equations

$x + y + z = 4$ $x^2 + y^2 + z^2 = 4$ $x^3 + y^3 + z^3 = 4$ Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex ...
15
votes
2answers
2k views

Why does the discriminant of a cubic polynomial being less than 0 indicate complex roots?

The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also ...