pls help with the above question
Answers
Answer:
(e) 1/18
Step-by-step explanation:
Given ---> α is a root of the equation
x² - 3x +1 = 0
To find ---> Value of
α³ / (α⁶ + 1)
Solution--->
ATQ
α is root of the equation
x² - 3 x + 1 = 0 ---------------(1)
Putting x =α in the equation (1)
α² - 3α + 1 = 0
From this we find two relations as follows
α² = 3α - 1
α² + 1 = 3α
Now we find value of
α⁶ + 1 = ( α² )³ + ( 1 )³
We have an identity
x³ + y³ = (x + y ) (x² + y² - xy )
Applying it here
= ( α² + 1 ) { (α² )² + ( 1 )² - α² (1 ) }
Putting α² = 3α -1 and α² + 1 = 3α
= (3 α ) {( 3α - 1 )² + 1 - (3α - 1 ) }
= (3α ) ( 9α² + 1 - 6α +1 - 3α + 1 )
= (3α ) (9α² - 9α + 3 )
= (3α ) {9 (3α - 1 ) - 9 α + 3 }
= (3α ) ( 27α - 9 - 9α + 3 )
= (3α ) ( 18α - 6 )
= (3α ) 6 ( 3α - 1 )
= 18 α ( 3α - 1 )
Now we find value of
α³ = α α²
= α ( 3α - 1 )
Now returning to original problem
α³ /( α⁶ + 1 ) = α (3α -1 ) / 18 α (3α -1 )
α ( 3α - 1 ) cancel out from numerator and denominator
α³ / (α⁶ - 1 ) = 1 / 18
e 1/18
#answerwithquality #bal