Question

What is the value of a^3 - b^3 if a/b + b/a = -1(a,b ≠ 0)?

Answer 1

Given :-

$\rm :\longmapsto\:\dfrac{a}{b} + \dfrac{b}{a} = - 1$

To Find :-

$\rm :\longmapsto\: {a}^{3} - {b}^{3}$

Identity Used :-

$\bf :\longmapsto\: {a}^{3} - {b}^{3} = (a - b)( {a}^{2} + ab + {b}^{2})$

Solution :-

Given that

$\rm :\longmapsto\:\dfrac{a}{b} + \dfrac{b}{a} = - 1$

$\rm :\longmapsto\:\dfrac{ {a}^{2} + {b}^{2} }{ab} = - 1$

$\rm :\longmapsto\: {a}^{2} + {b}^{2} = - ab$

$\bf\implies \: {a}^{2} + {b}^{2} + ab = 0 - - (1)$

Consider,

$\rm :\longmapsto\: {a}^{3} - {b}^{3}$

$\:  \: \rm  =  \:  \: (a - b)( {a}^{2} + {b}^{2} + ab)$

$\:  \: \rm  =  \:  \: (a - b) \times 0 \: \: \: \: \: \: \: \{using \: (1) \}$

$\:  \: \rm  =  \:  \: 0$

$\bf\implies \: {a}^{3} - {b}^{3 } = 0$

Additional Information :-

More Identities to know:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

• (a + b)² + (a - b)² = 2(a² + b²)

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)