Factorise $ab(x^{2} + y^{2}) + xy(a^{2} + b^{2})$.

Factorise $ab(x^{2} + y^{2}) + xy(a^{2} + b^{2})$.
  1. $(ax + by)(bx + ay)$
  2. $(ax - by)(bx - ay)$
  3. $(ax + by)(bx - ay)$
  4. $(a+b)(x+y)(ab+xy)$

Solution

Expand $(ax+by)(bx+ay) = abx^{2} + a^{2}xy + b^{2}xy + aby^{2} = ab(x^{2}+y^{2}) + xy(a^{2}+b^{2})$. So the factorisation is $(ax+by)(bx+ay)$.

Asked in: IMO

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