a: \(A=x^3-3x+3x^2y+3xy^2+y^3-3y\)
\(=\left(x^3+3x^2y+3xy^2+y^3\right)-3\left(x+y\right)\)
\(=\left(x+y\right)^3-3\left(x+y\right)\)
\(=0^3-3\cdot0=0\)
b: \(B=y^2-2x-9x^2-6x\)
\(=\left(y^2-9x^2\right)-2\left(y+3x\right)\)
\(=\left(y+3x\right)\left(y-3x\right)-2\left(y+3x\right)\)
\(=\left(y+3x\right)\left(y-3x-2\right)\)
\(=\left(y+3x\right)\left(y-3x-y+3x\right)=0\)
`A = x^3 - 3x + 3x^2 y + 3xy^2 + y^3 - 3y`
`= (x^3 + y^3) + (3x^2 y + 3xy^2) - (3x + 3y) `
`= (x+y)(x^2 - xy + y^2) + 3xy (x + y) - 3(x+y) `
`= (x+y) (x^2 - xy + y^2 + 3xy - 3)`
`= (x+y) (x^2 +2xy + y^2 - 3)`
`= (x+y) [(x+y)^2 - 3] `
`= 0 . (0^2 - 3) `
`= 0`
`B = y^2 - 2y - 9x^2 - 6x `
`= (y^2 - 9x^2) - (2y + 6x) `
`= (y-3x)(y+3x) - 2(y+3x)`
`= (y+3x)(y-3x-2) `
`= (y+3x)(2-2)`
`= (y+3x) . 0`
`= 0`
a)
\[ A = x^3 - 3x + 3x^2y + 3xy^2 + y^3 - 3y \]
\[
A = x^3 - 3x + 3x^2(-x) + 3x(-x)^2 + (-x)^3 - 3(-x)
\]
1. \( 3x^2(-x) = -3x^3 \)
2. \( 3x(-x)^2 = 3x(x^2) = 3x^3 \)
3. \( (-x)^3 = -x^3 \)
4. \( -3(-x) = 3x \)
\[
A = x^3 - 3x - 3x^3 + 3x^3 - x^3 + 3x
\]
\[
A = x^3 - 3x - x^3 + 3x = 0
\]
b)
\[
B = y^2 - 2y - 9x^2 - 6x
\]
\[
B = (3x + 2)^2 - 2(3x + 2) - 9x^2 - 6x
\]
1. \( (3x + 2)^2 = 9x^2 + 12x + 4 \)
2. \( -2(3x + 2) = -6x - 4 \)
\[
B = 9x^2 + 12x + 4 - 6x - 4 - 9x^2 - 6x
\]
\[
B = 9x^2 - 9x^2 + 12x - 6x - 6x + 4 - 4 = 0
\]
