Como fatorizar x ^ 7 + x ^ 5 + 1 (caso especial)?

Es obvio que no tiene un factor de la forma<img src="https://tex.z-dn.net/?f=%28x-a%29" /> donde<img src="https://tex.z-dn.net/?f=a%5Cin%5Cmathbb%20Q" />, entonces podemos pensar que tiene un factor de la forma<img src="https://tex.z-dn.net/?f=%28x%5E2%2Bax%2B1%29" /> y el otro<img src="https://tex.z-dn.net/?f=%28x%5E5%2Bbx%5E4%2Bcx%5E3%2Bdx%5E2%2Bex%2B1%29" />

Al multiplicarlos tenemos

<img src="https://tex.z-dn.net/?f=x%5E7%2B%28a%2Bb%29x%5E6%2B%28ab%2Bc%2B1%29x%5E5%2B%28ac%2Bb%2Bd%29x%5E4%2B%28ad%2Bc%2Be%29x%5E3%2B%5Ccdots%5C%5C%20%5C%5C%0A%5Chspace%2A%7B5cm%7D%5Ccdots%20%2B%28ae%2Bd%2B1%29x%5E2%2B%28a%2Be%29x%2B1%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctexttt%7BEntonces%20tenemos%3A%7D%0A%5Cbegin%7Bcases%7D%0Aa%2Bb%3D0%5C%5C%0Aab%2Bc%2B1%3D1%5C%5C%0Aac%2Bb%2Bd%3D0%5C%5C%20%0Aad%2Bc%2Be%3D0%5C%5C%0Aae%2Bd%2B1%3D0%5C%5C%0Aa%2Be%3D0%0A%5Cend%7Bcases%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctexttt%7BHaciendo%3A%20%7Da%3Dp%5Cto%20b%3D-p%5Cto%20e%3D-p%5Ctexttt%7B%20tenemos%7D%5C%5C%20%5C%5C%0A%5Cbegin%7Bcases%7D%0A-p%5E2%2Bc%3D0%5C%5C%0Apc-p%2Bd%3D0%5C%5C%20%0Apd%2Bc-p%3D0%5C%5C%0A-p%5E2%2Bd%2B1%3D0%5C%5C%0A%5Cend%7Bcases%7D%5C%5C%20%5C%5C%0Ac%3Dp%5E2%5Cwedge%20d%3Dp%5E2-1%5C%5C%20%5C%5C%0A%5Cbegin%7Bcases%7D%0Ap%5E3-p%2B%28p%5E2-1%29%3D0%5C%5C%20%0Ap%28p%5E2-1%29%2Bp%5E2-p%3D0%5C%5C%0A%5Cend%7Bcases%7D%5Cto%20p%3D1%0A" />

Luego

<img src="https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%0Aa%3D1%5C%5Cb%3D-1%5C%5Cc%3D1%5C%5Cd%3D0%5C%5Ce%3D-1%0A%5Cend%7Bcases%7D%5C%5C%20%5C%5C%20%5C%5C%20%0A%5Ctexttt%7BEntonces%7D%20%5C%5C%20%5C%5C%0Ax%5E7%2Bx%5E5%2B1%3D%28x%5E2%2Bx%2B1%29%28x%5E5-x%5E4%2Bx%5E3-x%2B1%29" />.