Divide $x^{4 a} + x^{2 a} y^{2 b} + y^{4 b}$ by $x^{2 a} + x^{a} y^{b} + y^{2 b}$ [4 MARKS]

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Solution

Concept: 1 Mark
Application: 3 Marks

We have,

$x^{4 a} + x^{2 a} y^{2 b} + y^{4 b} = (x^{2 a})^{2} + x^{2 a} y^{2 b} + (y^{2 b})^{2}$

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

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

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

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

$∴ \frac{x^{4 a} + x^{2 a} y^{2 b} + y^{4 b}}{x^{2 a} + x^{a} y^{b} + y^{2 b}}$

$= \frac{(x^{2 a} + x^{a} y^{b} + y^{2 b}) (x^{2 a} - x^{a} y^{b} + y^{2 b})}{(x^{2 a} + x^{a} y^{b} + y^{2 b})}$

$= x^{2 a} - x^{a} y^{b} + y^{2 b}$

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