Factorise: $x (y^{2} - z^{2}) + y (z^{2} - x^{2}) + z (x^{2} - y^{2})$ .

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Solution

$x (y^{2} - z^{2}) + y (z^{2} - x^{2}) + z (x^{2} - y^{2}) = x y^{2} - x z^{2} + y z^{2} - y x^{2} + z x^{2} - z y^{2}$
Arrange the terms in descending order of power of x.
$= - x^{2} y + z x^{2} + x y^{2} - x z^{2} - y^{z} + y z^{2}$
$= - x^{2} (y - z) + x (y^{2} - z^{2}) - y z (y - z)$
$= - x^{2} (y - z) + x (y - z) (y + z) - y z (y - z)$
$= (y - z) [- x^{2} + x (y + z) - y z]$
$(y - z) [- x^{2} + x y + x z - y z]$

Arrange the term in bracket in descending order of the power of y.
$= (y - z) [x y - y z - x^{2} + x z]$
$= (y - z) [- y (- x + z) + x (- x + z)]$
$= (x - y) (y - z) (z - x)$ .

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Write the following by using ∑ notation

(1) ab(a + b) + bc(b + c) + ca(c + a)

(2) a² + b² + c² − ab − bc − ca

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(4) ab² − ac² − a²b + bc² − cb² + ca²

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