Divide $2 x^{3 a} - 6 x^{2 a} y^{a} + 6 x^{a} y^{2 a} - 2 y^{3 a}$ by $2 x^{a} - 2 y^{a}$ [4 MARKS]

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Solution

Concept: 1 Mark
Application: 3 Marks

$2 x^{3 a} - 6 x^{2 a} y^{a} + 6 x^{a} y^{2 a} - 2 y^{3 a}$

$= 2 (x^{3 a} - 3 x^{2 a} y^{a} + 3 x^{a} y^{2 a} - y^{3 a})$

$= 2 {(x^{a})^{3} - (y^{a})^{3} - 3 x^{a} y^{a} (x^{a} - y^{a})} [∵ x^{3 a} = (x^{a})^{3} a n d y^{3 a} = (y^{a})^{3}]$

$U s i n g t h e i d e n t i t y a^{3} - b^{3} - 3 a b (a - b) = (a - b)^{3}$

$= 2 (x^{a} - y^{a})^{3}$

$2 x^{a} - 2 y^{a} = 2 (x^{a} - y^{a})$

$∴ \frac{2 x^{3 a} - 6 x^{2 a} y^{a} + 6 x^{a} y^{2 a} - 2 y^{3 a}}{2 x^{a} - 2 y^{a}} = \frac{2 (x^{a} - y^{a})^{3}}{2 (x^{a} - y^{a})}$

$= (x^{a} - y^{a})^{2}$

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