Question

x square - 7 x + 2 equals to zero and alpha beta are the roots of the equation then find Alpha ^10-beta^10+2(alpha^8-beta^8) divided by alpha^9-beta^9​

Answer 1

Given :  x² - 7x  + 2  = 0  , alpha α  , beta βare the roots of the equation

To find : (( α¹⁰ - β¹⁰) + 2 (α⁸ - β⁸) ) /( α⁹ - β⁹)

Solution:

x² - 7x  + 2  = 0

α & β  are roots

=> α + β = -(-7)  = 7

 αβ = 2

(( α¹⁰ - β¹⁰) + 2 (α⁸ - β⁸) ) /( α⁹ - β⁹)

putting 2 =  αβ

= (( α¹⁰ - β¹⁰) + αβ (α⁸ - β⁸) ) /( α⁹ - β⁹)

= (  α¹⁰ - β¹⁰  + α⁹β  - αβ⁹) ) /( α⁹ - β⁹)

=  (  α¹⁰ + α⁹β   - β¹⁰ - αβ⁹) ) /( α⁹ - β⁹)

= (α⁹(α + β) - β⁹( β + α) ) / ( α⁹ - β⁹)

= (α⁹(α + β) - β⁹( α  + β) ) / ( α⁹ - β⁹)

= ((α⁹ - β⁹)( α  + β) ) / ( α⁹ - β⁹)

= α  + β

= 7

(( α¹⁰ - β¹⁰) + 2 (α⁸ - β⁸) ) /( α⁹ - β⁹) = 7

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