a) \(x^4+4y^4\)
\(=\left(x^2\right)+\left(2y^2\right)^2\)
\(=\left(x^2\right)^2+2.x^2.2y^2+\left(2y^2\right)^2-2.x^2.2y^2\)
\(=\left(x^2+2y^2\right)^2-4x^2y^2\)
\(=\left(x^2+2y^2\right)^2-\left(2xy\right)^2\)
\(=\left(x^2+2y^2-2xy\right)\left(x^2+2y^2+2xy\right)\)
b) \(x^4+y^4\)
\(=\left(x^2\right)^2+2x^2y^2+\left(y^2\right)^2-2x^2y^2\)
\(=\left(x^2+y^2\right)^2-\left(\sqrt{2}xy\right)^2\)
\(=\left(x^2+y^2-\sqrt{2}xy\right)\left(x^2+y^2+\sqrt{2}xy\right)\)
c) \(4x^4+y^4\)
\(=\left(2x^2\right)^2+\left(y^2\right)^2\)
\(=\left(2x^2\right)^2++2.2x^2.y^2+\left(y^2\right)^2-2.2x^2.y^2\)
\(=\left(2x^2+y^2\right)^2-4x^2y^2\)
\(=\left(2x^2+y^2\right)^2-\left(2xy\right)^2\)
\(=\left(2x^2+y^2\right)^2-\left(\sqrt{2}xy\right)^2\)
\(=\left(2x^2+y^2-\sqrt{2}xy\right)\left(2x^2+y^2+\sqrt{2}xy\right)\)
d) \(x^4+324\)
\(=\left(x^2\right)^2+18^2\)
\(=\left(x^2\right)^2+2.x^2.18+18^2-2.x^2.18\)
\(=\left(x^2+18\right)^2-36x^2\)
\(=\left(x^2+18\right)^2-\left(6x\right)^2\)
\(=\left(x^2+18-6x\right)\left(x^2+18+6x\right)\)
e) \(x^5+x+1\)
\(=x^5+x^4+x^3-x^4-x^3-x^2+x^2+x+1\)
\(=\left(x^5+x^4+x^3\right)-\left(x^4+x^3+x^2\right)+\left(x^2+x+1\right)\)
\(=x^3\left(x^2+x+1\right)-x^2\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^3-x^2+1\right)\)
f) \(x^5-x^4-1\)
\(=x^5-x^3-x^2-x^4+x^2+x+x^3-x-1\)
\(=\left(x^5-x^3-x^2\right)-\left(x^4-x^2-x\right)+\left(x^3-x-1\right)\)
\(=x^2\left(x^3-x-1\right)-x\left(x^3-x-1\right)+\left(x^3-x-1\right)\)
\(=\left(x^3-x-1\right)\left(x^2-x+1\right)\)
\(a)x^4+4y^4\)
\(=\left(x^2\right)^2+\left(2y^2\right)^2\)
\(=\left(x^2\right)^2+4x^2y^2+\left(2y^2\right)^2-4x^2y^2\)
\(=\left(x^2+y^2\right)^2-\left(2xy\right)^2\)
\(=\left(x^2+2y^2-2xy\right)\left(x^2+2y^2+2xy\right)\)