Equation of one variable

Question

Solve the following equation. $$(8-x)\cdot(x^2-2x+16)^2+2x^4\cdot(x^2-2x+16)=16x^7,$$ where $x\in \mathbb{R}$

I already know that we need to prove $x=2$, but don't know how to show it... I'd be grateful for some hints or solutions ;)

Answer 1

hint: your equation can be written as $$- \left( x-2 \right) \left( 16\,{x}^{6}+30\,{x}^{5}+65\,{x}^{4}+86\,{ x}^{3}+240\,{x}^{2}+128\,x+1024 \right) =0$$

Answer 2

It's $$(x-2)(16x^6+30x^5+65x^4+86x^3+240x^2+128x+1024)=0$$ and since $$16x^6+30x^5+65x^4+86x^3+240x^2+128x+1024=$$ $$=(16x^6+30x^5+15x^4)+(50x^4+86x^3+37x^2)+(203x^2+128x+1024)>0,$$ we get the answer: $\{2\}$.