InterviewSolution
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InterviewSolution
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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` |
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Answer» 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` |
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