Show that the polynomial `x^(4p)+x^(4q+1)+x^(4r+2)+x^(4s+3)` is divisible by `x^3+x^2+x+1, w h e r ep ,q ,r ,s in ndot`

Explanation

Let `f(x) = x ^(4p) + x^(4q+1) + x^(4r +2) + x^(4s + 3)`. Now
`x^(3) + x^(2) + x + 1 = (x^(2) + 1) (x+1)`
` = (x^(2) - i^(2))(x + 1)`
`= (x + i) (x- i)(x +1)`
` f (i) = = i^(4p) + i^(4q+1) + i^(4r + 2) + i^(4s + 3)`
`= 1 + i^(1) + i^(2) + ^(3)`
`= 1 + i -1-i=0`
`f (-i) = (-i)^(4p+1) +(-1)^(4r+2) + (-i)^(4s +3)`
` = 1 + (-i)^(1) +i= 0`
`f (-1) = (-1)^(4p) + (-1)^(4q+1) + (-1) ^(4r+2)+ (-1)^(4s+3) = 0`
Thus, by factor therem f(x) is divisible by `x^(3) + x^(2) + 1`