Question

if alfa and beta are the zeroes of p (x)=x^2+5x+2 find the value of alfa^5+beta^5​

Answer 1

Answer:

- 1975

Step-by-step explanation:

Given-----> P ( x ) = ( x² + 5x + 2 ) , and α and β are zeroes of given expression .

To find----> Value of ( α⁵ + β⁵ ) .

Solution-----> ATQ,

P ( x ) = x² + 5x + 2

We know that ,

Sum of zeroes = - b / a

=> α + β = - 5 / 1

=> α + β = - 5

Product of zeroes = c / a

=> α β = 2 / 1

=> α β = 2

Now , we know that ,

( α + β )² = α² + β² + 2αβ

=> ( - 5 )² = ( α² + β² ) + 2 ( 2 )

=> 25 = ( α² + β² ) + 4

=> 25 - 4 = ( α² + β² )

=> ( α² + β² ) = 21

Now we know that,

( α + β )³ = α³ + β³ + 3αβ ( α + β )

=> ( - 5 )³ = ( α³ + β³ ) + 3 ( 2 ) ( - 5 )

=> - 125 = ( α³ + β³ ) - 30

=> ( α³ + β³ ) = - 125 + 30

=> ( α³ + β³ ) = - 95

Now,

( α³ + β³ ) ( α² + β² ) = α⁵ + α³ β² + β³α² + β⁵

= ( α⁵ + β⁵ ) + ( α³ β² + β³ α² )

=> ( α³ + β³ ) ( α² + β² ) = ( α⁵ + β⁵ ) + α² β² ( α + β )

=> ( -95 ) ( 21 ) = ( α⁵ + β⁵ ) + ( αβ )² ( - 5 )

=> - 1995 = ( α⁵ + β⁵ ) - 5 ( 2 )²

=> - 1995 = ( α⁵ + β⁵ ) - 20

=> - 1995 + 20 = ( α⁵ + β⁵ )

=> ( α⁵ + β⁵ ) = - 1975