a) 125x3 + y6 = (5x)3 + (y2)3 = (5x + y2)(25x2 - 5xy2 + y4)
b) x³ + y³ + z³ - 3xyz
= (x+y)³ - 3xy(x-y) + z³ - 3xyz
= (x+y)³ + z³ - 3xyz - 3xy(x-y)
= (x+y+x)[(x+y)2 - (x+y).z + z2] - 3xy (z+x+y)
= (x+y+x)[(x+y)2 - (x+y).z + z2 -3xy]
= (x+y+x)[x2 + y2 - xz - yz + z2 -3xy]
c) (x2 - 8)2 + 36 = x4 - 16x2 + 100 = x4 + 20x2 - 36x2 + 100
= (x2 + 10)2 - (6x)2
= (x2 + 10 - 6x)(x2 + 10 + 6x)
a) \(125x^3+y^6\)\(=\left(5x\right)^3+\left(y^2\right)^3\)
\(=\left(5x+y^2\right)\left(25x^2-5xy^2+y^4\right)\)
b) \(x^3+y^3+z^3-3xyz\)
\(=\left(x+y\right)^3+z^3-3xy\left(x+y\right)-3xyz\)
\(=\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2\right]-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2\right)-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)\)
c) \(\left(x^2-8\right)^2-36=\left(x^2-8\right)^2-6^6\)
\(=\left(x^2-8-6\right)\left(x^2-8+6\right)\)
\(=\left(x^2-14\right)\left(x^2-2\right)\)
\(=\left(x-\sqrt{14}\right)\left(x+\sqrt{14}\right)\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right)\)
