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InterviewSolution
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Find the remainder when `x^999` is divided by `x^2-4x + 3`. |
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Answer» Let `q(x)` and mx + n be the quotient and the remainder respectively when `x^(999)` is divided by `x^(2)-4x+3`. `:. x^(999)=(x^(2)-4x+3)q(x)+mx+n`. If ` x= 1` , `x^(999)=(1-4+3)q(x)+m(1)+n` `rArr1=0xxq(x)+m+n` `rArrm+n=1` (1) If ` x= 3`, `3^(999)=(3^(2)-4(3)+3)q(x)+3m+n` `rArr3^(999)=0xxq(x)+3m+n` `rArr3m+n=3^(999)` (2) subtracting Eq . (1) from Eq . (2) , we get `2m=3^(999)-1` `m=(1)/(2)(3^(999)-1)` Substituting m in Eq . (1) , we have `n=1-(1)/(2)(3^(999)-1)=1-(1)/(2)3^(999)+(1)/(2)=(3)/(2)-(1)/(2)3^(999)` `n=(3)/(2)(1-3^(999))` `:.` The required remainder is `(1)/(2)(3^(999)-1)x+(3)/(2)(1-3^(999))`. |
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