Consider the expression .

MATHEMATICS · Multi-part · AoPS · Mathematics · Special Factorizations · Special Factorizations

Problem

Consider the expression

E=x^{4}-4x^{2}y^{2}+y^{4}

. (i) Show that

E=(x^{2}-2xy+y^{2})(x^{2}+2xy+y^{2})

. (ii) Hence factorise

E

completely into linear factors. (iii) Verify that the factorised form obtained in part (ii) can also be written as

(x-y)^{2}(x+y)^{2}

. (iv) If

x

and

y

are integers satisfying

E=0

, find all possible ordered pairs

(x,y)