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## Friday, February 17, 2012

### Polynomial exercise - Solution.

In a recent post I proposed the following exercise.

Let $$x^3 + bx^2 + cx + d$$ be a polynomial with coefficients in $\mathbf{Q}$. We ask which condition(s) $b,c,d$ must satisfy in order that one ( any ) root be the average of the other two roots?

Solution.

Let the roots of $x^3 + bx^2 + cx + d = 0$ be $\alpha_1, \alpha_2, \alpha_3$. Clearly, the roots must satisfy the following equations:
\begin{align*}
\alpha_1=& \frac{1}{2}\alpha_2 + \frac{1}{2}\alpha_3 \\
\alpha_2=& \frac{1}{2}\alpha_1 + \frac{1}{2}\alpha_3 \\
\alpha_3=& \frac{1}{2}\alpha_1 + \frac{1}{2}\alpha_2 \\
\end{align*}
This is a rank $2$ system of linear equations with solution $( \alpha_1, \alpha_2, \alpha_3) = \lambda (1,1,1)$, and implies that the equation has three equal roots and can thus be written as follows:$$(x-\alpha)^3 = x^3-3\alpha x^2 + 3\alpha^2 x - \alpha^3$$.
So if the root is $\alpha$ then $b=-3\alpha, c=3\alpha^2$ and $d=-\alpha^3$.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)