`If (1-x+x^(2))^(4)=1+P_(1)x+P_(2)x^(2)+P_(3)x^(3)+…….+P_(8)x^(8),` then prove that :
`P_(2)+P_(4)+_(6)+P_(8)=40 " and "P_(1)+P_(3)+P_(5)+P_(7)=-40`

Answer

N/a

Explanation

Givent that
`(1-x+x^(2))^(4)=1+P_(1)x+P_(2)x^(2)+P_(3)x^(3)+......+P_(8)x^(8) " "......(i)`
Putting x=1 on both sides
`1=1+P_(1)+P_(2)+P_(3)+……+P_(8) " "……(2)`
Putting x=-1 in eq . (1)
`3^(4)=1-P_(1)+P_(2)-P_(3)…..+P_(8) " "......(3)`
Adding eqs. (2) and (3)
`82 =2 (1+P_(2)+P_(4)+P_(6)+P_(8))`
`rarr " "P_(2)+P_(4)+P_(6)+P_(8)=40`
Subtract eq. (3) from eq. (2)
`-80 =2 (P_(1)+P_(3)+P_(5)+P_(7))`
`rArr" "P_(1)+_P_(3)+P_(5)P_(7)=-40`
Therefore `P_(2)+P_(4)+P_(6)+P_(8)=40`
`" and " P_(1)+P_(3)+P_(5)+P_(7) =- 40` Hence Proved.