Thursday, December 29, 2016

algebra precalculus - Find remainder of division of x3 by x2x+1




I am stuck at my exam practice here.





The remainder of the division of x3 by x2x+1 is ..... and that of x2007 by x2x+1 is .....




I tried the polynomial remainder theorem but I am not sure if I did it correctly.



By factor theorem definition, provided by Wikipedia,





the remainder of the division of a polynomial f(x) by a linear polynomial xr is equal to f(r).




So I attempted to find r by factorizing x2x+1 first but I got the complex form x=1±3i2=r.



f(r) is then (1+3i2)3 or (13i2)3 which do not sound right.



However, the answer key provided is 1 for the first question and also 1 for the second one. Please help.


Answer



Since x3+1=(x+1)(x2x+1) so x3=(x+1)(x2x+1)1 the answer is 1.




Similarly for x3n+1=(x3+1)((x3)n1(x3)n2+...(x3)+1)q(x)=(x+1)(x2x+1)q(x)



so the answer is again 1.


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