Question

If \alphaα and \betaβ are the roots of the equation x^{4}-x^{3}+1=0x
4
−x
3
+1=0 then the value of \frac{\alpha^{3}(1-\alpha)}{\beta^{3}(1-\beta)}=
β
3
(1−β)
α
3
(1−α)
​ =​

Answer 1

Answer:    α  & β  are roots of equation  x⁴ -  x³   +  1  = 0

To find : Value of    $\frac{\alpha^{3}(1-\alpha)}{\beta^{3}(1-\beta)}$     α³(1 - α) / β³(1 - β)  =

Solution:

x⁴ -  x³   +  1  = 0

α  is a root

=> α⁴ -  α³   +  1  = 0

=> α³(α - 1) + 1 = 0

=> α³(α - 1) = - 1

multiplying by - 1 both sides

=> α³(1 - α)  =  1

x⁴ -  x³   +  1  = 0

β  is a root

=> β⁴ -  β³   +  1  = 0

=> β³(β - 1) + 1 = 0

=> β³(β - 1) = - 1

multiplying by - 1 both sides

=> β³(1 - β)  =  1

α³(1 - α) / β³(1 - β)  =  1/1

=> α³(1 - α) / β³(1 - β)  = 1

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