Question

Factorise: $x^6-729$.

Answer 1

Factorize the given polynomial

Given: $x^6-729={\left(x^2\right)}^3-{\left(3^2\right)}^3$

Use the identity $a^3-b^3=(a-b)(a^2+b^2+ab)$

$x^6-729=(x^2-3^2)(x^4+3^4+x^2\times 3^2)$

Use the identity $a^2-b^2=(a-b)(a+b)$

$x^6-729=(x-3)(x+3)(x^4+81+9x^2)$

Hence the factorized form of the polynomial $x^6-729$ is $(x-3)(x+3)$$(x^4+81+9x^2)$.