Question

Разложи на множители: (y14+x14)²−(y14−x14)²−y²x²

Answer 1

$(y^{14}+x^{14})^2-(y^{14}-x^{14})^2-y^2x^2=\\\\=y^{28}+2x^{14}y^{14}+x^{28}-(y^{28}-2x^{14}y^{14}+x^{28})-x^2y^2=\\\\=4x^{14}y^{14}-x^2y^2=(2x^7y^7)^2-(xy)^2=\\\\=(2x^7y^7-xy)(2x^7y^7+xy)=xy\cdot xy\cdot (2x^6y^6-1)(2x^6y^6+1)=\\\\=x^2y^2\cdot (\sqrt2x^3y^3-1)(\sqrt2x^3y^3+1)(2x^6y^6+1)=\\\\=x^2y^2\, (\sqrt[6]2xy-1)(\sqrt[3]2x^2y^2+\sqrt[6]2xy+1) (\sqrt[6]2xy+1)(\sqrt[3]2x^2y^2-\sqrt[6]2xy+1)\times \\\\\times (\sqrt[3]2x^2y^2+1)(\sqrt[3]4x^4y^4-\sqrt[3]2x^2y^2+1)$