It is given polynomial `P(x)=(xsina+cosa)^n` for natural number n. Finding the remaining when `P(x)` divide to `(x^2+1)`

 It is given polynomial `P(x)=(xsina+cosa)^n` for natural number n.

Finding the remaining when `P(x)` divide to `(x^2+1)`

Solution

Let `R(x)` be the remainder of division between `P(x)` and `(x^2+1)`

• If `n=1` then `P(x)=xsina+cosa`

Therefore, the remainder `R(x)=xsina+cosa`

• If `n` ≥ `2` then `P(x)=(x^2+1)Q(x)+R(x)`

which `Q(x)` is the result and `R(x)=Ax+B`

Then, `(xsina+cosa)^n=(x^2+1)Q(x)+Ax+B`

If `x=i` then `cosna+isinna=B+iA`

Therefore, `A=sin(na) ;B=cos(na)`

Hence, the remainder is `R(x)=sin(na)x+cons(na)` .

Solution by: Thin Sokkean