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let-give-P-n-x-x-1-2n-x-2-n-1-and-Q-x-x-2-3x-2-find-R-x-P-n-x-R-x-Q-x-

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Question Number 28312 by abdo imad last updated on 23/Jan/18
let give P_n (x)=(x+1)^(2n) +(x+2)^n −1 and Q(x)= x^2 +3x +2 find R(x) /P_n (x)=R(x) Q(x) .
letgivePn(x)=(x+1)2n+(x+2)n−1andQ(x)=x2+3x+2findR(x)/Pn(x)=R(x)Q(x).
Answered by sma3l2996 last updated on 24/Jan/18
P(x)=R(x)Q(x) so R(x)=((P(x))/(Q(x))) R(x)=(((x+1)^(2n) )/(x^2 +3x+2))+(((x+2)^n )/(x^2 +3x+2))−(1/(x^2 +3x+2)) we have x^2 +3x+2=(x+1)(x+2) so R(x)=(((x+1)^(2n−1) )/(x+2))+(((x+2)^(n−1) )/(x+1))−(1/((x+1)(x+2))) let′s put t=x+1 so x+2=t+1 so t^(2n−1) =(t^(2n−2) −t^(2n−3) +t^(2n−4) −...+1)(t+1)−1 so (x+1)^(2n−1) =(Σ_(k=2) ^(2n) (−1)^k (x+1)^(2n−k) )(x+2)−1 (((x+1)^(2n−1) )/(x+2))=Σ_(k=2) ^(2n) (−1)^k (x+1)^(2n−k) −(1/(x+2)) samething for (((x+2)^(n−1) )/(x+1))=Σ_(k=2) ^n (x+2)^(n−k) +(1/(x+1)) so R(x)=Σ_(k=2) ^(2n) (−1)^k (x+1)^(2n−k) +Σ_(k=2) ^n (x+2)^(n−k) +(1/(x+1))−(1/(x+2))−(1/((x+1)(x+2))) (1/((x+2)(x+1)))=(1/(x+1))−(1/(x+2)) so R(x)=Σ_(k=2) ^n (x+2)^(n−k) +Σ_(k=2) ^(2n) (−1)^k (x+1)^(2n−k)
P(x)=R(x)Q(x)soR(x)=P(x)Q(x)R(x)=(x+1)2nx2+3x+2+(x+2)nx2+3x+2−1x2+3x+2wehavex2+3x+2=(x+1)(x+2)soR(x)=(x+1)2n−1x+2+(x+2)n−1x+1−1(x+1)(x+2)let′sputt=x+1sox+2=t+1sot2n−1=(t2n−2−t2n−3+t2n−4−…+1)(t+1)−1so(x+1)2n−1=(∑2nk=2(−1)k(x+1)2n−k)(x+2)−1(x+1)2n−1x+2=∑2nk=2(−1)k(x+1)2n−k−1x+2samethingfor(x+2)n−1x+1=∑nk=2(x+2)n−k+1x+1soR(x)=∑2nk=2(−1)k(x+1)2n−k+∑nk=2(x+2)n−k+1x+1−1x+2−1(x+1)(x+2)1(x+2)(x+1)=1x+1−1x+2soR(x)=∑nk=2(x+2)n−k+∑2nk=2(−1)k(x+1)2n−k

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