Iterated factorisation with grouping and difference of squares

The techniques chain together: after a partial grouping, the factors obtained may in turn be differences of squares.

Example — Factorise $x^4-4x^2y^2+4y^6-x^2y^4$

Step 1 — Partial grouping (Venn diagram).

The partial grouping brings out the common factor $(x^2-4y^2)$.

Step 2 — Difference of squares (method of forms). Both factors are differences of squares:

• $x^2-4y^2$: with $\bigcirc=x$ and $\square=2y$ we get $(x+2y)(x-2y)$.
• $x^2-y^4$: with $\bigcirc=x$ and $\square=y^2$ we get $(x+y^2)(x-y^2)$.

Final result: $x^4-4x^2y^2+4y^6-x^2y^4 = \boxed{(x+2y)(x-2y)(x+y^2)(x-y^2)}$

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Topics: Algebraic calculation
Concepts: Difference of squares · Partial grouping · Factorisation
Skills: Factorising