Sum and difference of cubes

Statement

Factor: (a) $b^3 + 64$; (b) $8x^3 + 27y^3$; (c) $2r^3 - 54$; (d) $1000 - x^6 y^3$.

Solution

Use the identities $A^3 + B^3 = (A + B)(A^2 - AB + B^2)$ and $A^3 - B^3 = (A - B)(A^2 + AB + B^2)$.

(a) $b^3 + 64 = b^3 + 4^3 = (b + 4)(b^2 - 4b + 16).$

(b) $8x^3 + 27y^3 = (2x)^3 + (3y)^3 = (2x + 3y)(4x^2 - 6xy + 9y^2).$

(c) First pull out $2$: $2r^3 - 54 = 2(r^3 - 27) = 2(r - 3)(r^2 + 3r + 9).$

(d) $1000 - x^6 y^3 = 10^3 - (x^2 y)^3 = (10 - x^2 y)(100 + 10 x^2 y + x^4 y^2).$

$\boxed{(b+4)(b^2-4b+16)\ ;\ (2x+3y)(4x^2-6xy+9y^2)\ ;\ 2(r-3)(r^2+3r+9)\ ;\ (10-x^2y)(100+10x^2y+x^4y^2)}$