Polynomial $x^3-2x^2-3x-4=0$

Question

Let $\alpha,\beta,\gamma$ be three distinct roots of the polynomial $x^3-2x^2-3x-4=0$. Then find $$\frac{\alpha^6-\beta^6}{\alpha-\beta}+\frac{\beta^6-\gamma^6}{\beta-\gamma}+\frac{\gamma^6-\alpha^6}{\gamma-\alpha}.$$ I tried to solve with Vieta's theorem. We have $$\begin{align} \alpha+\beta+\gamma &= 2, \\ \alpha\beta+\beta\gamma+\gamma\alpha &= -3, \\ \alpha\beta\gamma &= 4. \end{align}$$ For example, $\alpha^2+\beta^2+\gamma^2=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\beta\gamma+\gamma\alpha)=10$ and similarly, we can find $\alpha^3+\beta^3+\gamma^3$...

But it has very long and messy solution. Can anyone help me?

Answer 1

By long division, the remainder of $x^6$, when divided by $x^3-2x^2-3x-4$, is $77x^2+100x+96$. So we know that $$ \alpha^6=77\alpha^2+100\alpha+96 $$ and the same with other roots. Therefore you are looking at the sum $$ \begin{aligned} S&=\frac{77(\alpha^2-\beta^2)+100(\alpha-\beta)}{\alpha-\beta}+\text{cyclic}\\ &=77(\alpha+\beta)+100+\text{cyclic}\\ &=154(\alpha+\beta+\gamma)+300\\ &=608 \end{aligned} $$ by the (Vieta) relations you have.

Answer 2

Let $\alpha+\beta+\gamma=3u$, $\alpha\beta+\alpha\gamma+\beta\gamma=3v^2$ and $\alpha\beta\gamma=w^3$.

Thus, $$\sum_{cyc}\frac{\alpha^6-\beta^6}{\alpha-\beta}=\sum_{cyc}(2\alpha^5+\alpha^4\beta+\alpha^4\gamma+\alpha^3\beta^2+\alpha^3\gamma^2)=$$ $$=2(243u^5-405u^3v^2+135uv^4+45u^2w^3-15v^2w^3)+$$ $$+81u^3v^2-81uv^4-9u^2w^3+15v^2w^3+$$ $$+27uv^4-18u^2w^3-3v^2w^3=$$ $$=9(54u^5-81u^3v^2+7u^2w^3+24uv^4-2v^2w^3).$$ Now, use your work.

I got $608$.

Answer 3

If $\alpha,\beta,\gamma$ satisfy $x^6=ax^2+bx+c$, then the first term is $a(\alpha+\beta)+b$, and we have similar expressions for the others, so the desired sum is $2a(\alpha+\beta+\gamma)+3b=4a+3b$. To find $a$ and $b$, square $x^3$ and reduce modulo the given polynomial.

Answer 4

The final answer I got is 608. The answer for this question is provided in the given pdf By Vieta's theorem and using some special algebraic identities that are prevelant in India.. Click on the link .THE whole values are achieved by some easy calculations