square the last term (the sign \( a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right) \) The sum and difference of the cubes always factorise Exercise 2: Factorise a) \( x^{3}-y^{3} \) b) \( x^{3}+y^{3} \) c) \( 8 x^{3}+y^{3} \) d) \( 8 a^{3}-64 b^{3} \) e) \( (x-y)^{3}-64 x^{3} \) f) \( 54 x^{3}-16 \) g) \( 54 x^{5} y-250 x^{2} y \) h) \( 2 a x^{3}+3 b x^{3}-2 a y^{3}-3 b y^{3} \) (e)

The sum and difference of cubes are classic examples of algebraic identities that allow us to break down complex polynomial expressions into simpler factors. For \(x^3 - y^3\), you can factor it as \((x - y)(x^2 + xy + y^2)\), while \(x^3 + y^3\) transforms into \((x + y)(x^2 - xy + y^2)\). Knowing these identities can save you time and effort in solving equations! When tackling problems like these, keep an eye out for common errors such as forgetting to apply the correct signs in your factorization or neglecting the square terms. A prevalent mistake is getting the binomial coefficients wrong in the factorizations, especially in \( (x + y)^3 \) versus \( (x - y)^3 \). Double-checking your work and going through the steps one at a time can help you spot these blunders before they snowball into bigger issues!