Question
Factorize: $x^4 + x^2y^2 + y^4$
✓
Answer
$x^4 + x^2y^2 + y^4$
Adding $x^2y^2$ and subtracting $x^2y^2$ to the given equation
$= x^4 + x^2y^2 + y^4 + x^2y^2 - x^2y^2$
$= x^4 + 2x^2y^{2 }+ y^4 - x^2y^2$
$= (x^2)^2 + 2 \times x^2 \times y^2 + (y^2)^2 - (xy)^2$
Using the identity$ (p + q)^2 = p^2 + q^2 + 2pq$
$= (x^2 + y^2)^2 - (xy)^2$
Using the identity $p^2 - q^2 = (p + q)(p - q)$
$= (x^2 + y^2 + xy)(x^2 + y^2 - xy)$
$\therefore x^4 + x^2y^2 + y^4$
$ = (x^2 + y^{2 }+ xy)(x^2 + y^2 - xy)$
Adding $x^2y^2$ and subtracting $x^2y^2$ to the given equation
$= x^4 + x^2y^2 + y^4 + x^2y^2 - x^2y^2$
$= x^4 + 2x^2y^{2 }+ y^4 - x^2y^2$
$= (x^2)^2 + 2 \times x^2 \times y^2 + (y^2)^2 - (xy)^2$
Using the identity$ (p + q)^2 = p^2 + q^2 + 2pq$
$= (x^2 + y^2)^2 - (xy)^2$
Using the identity $p^2 - q^2 = (p + q)(p - q)$
$= (x^2 + y^2 + xy)(x^2 + y^2 - xy)$
$\therefore x^4 + x^2y^2 + y^4$
$ = (x^2 + y^{2 }+ xy)(x^2 + y^2 - xy)$
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