Simplify this expression: $25x^2 + 30x^3 - 35x^6$

Question

I was thinking about simplifying this expression:

$$25x^2 + 30x^3 - 35x^6$$

Note that I don't mean just factoring $5x^2$ from it!

Answer 1

Your expression factors as $\frac{-5}{7}x^2a(x)b(x)c(x)$, where $a(x)$, $b(x)$, and $c(x)$ are, respectively,

$$\left(\!\!x\! -\!\! \sqrt{\!\sqrt[3]{\left(81\!+\!\sqrt{48561}\right)^2}\! -\! 10\sqrt[3]{42}}\!+\!\frac{\!\!\!\!\!\sqrt{\!10\sqrt[3]{\!42}\! -\!\! \sqrt[3]{\!\left(81\!+\!\sqrt{48561}\right)^2}\!+\!\!18\sqrt{\!\!\frac{2\left(81+\sqrt{48561}\right)}{\!\sqrt[3]{\!\left(81+\sqrt{48561}\right)^2}\! - 10\sqrt[3]{42}}}}}{\sqrt[6]{2^5}\sqrt[3]{21}\sqrt[6]{81+\sqrt{48561}}}\!\!\right)$$ and $$\left(\!\!x\! -\!\! \sqrt{\!\!\sqrt[3]{\!\left(\!81\!\!+\!\!\sqrt{\!48561}\!\right)^2}\!\! -\!\! 10\sqrt[3]{42}}\!-\!\!\frac{\!\!\!\!\!\sqrt{\!10\sqrt[3]{\!42} \!-\!\! \sqrt[3]{\!\!\left(81\!\!+\!\!\sqrt{\!48561}\right)^2}\!\!+\!\!18\sqrt{\!\!\frac{2\left(81+\sqrt{48561}\right)}{\!\sqrt[3]{\!\left(81+\sqrt{48561}\right)^2} - 10\sqrt[3]{42}}}}}{\sqrt[6]{2^5}\sqrt[3]{21}\sqrt[6]{81+\sqrt{48561}}}\right)$$

and $$\left(x^2 -\alpha x+\beta\right),$$

where $$\alpha=2\sqrt{\sqrt[3]{\left(81+\sqrt{48561}\right)^2}-10\sqrt[3]{41}}$$

and $$\beta\!=\!\!\sqrt[3]{\!\left(81\!\!+\!\!\sqrt{\!48561}\right)^2}\!\!-\!\!10\sqrt[3]{42}\! +\!\! \frac{\!\!\!\sqrt[3]{\!\left(81\!\!+\!\!\sqrt{\!48561}\right)^2}\!\!-\!\!10\sqrt[3]{42}\!+\!\!18\sqrt{\!\!\frac{2\left(81+\sqrt{48561}\right)}{\!\sqrt[3]{\left(81+\sqrt{48561}\right)^2}-10\sqrt[3]{42}}}}{\sqrt[3]{2^5}\sqrt[3]{21^2}\sqrt[3]{81+\sqrt{48561}}}.$$