I am stuck at my exam practice here.
The remainder of the division of x3 by x2−x+1 is ..... and that of x2007 by x2−x+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 x−r is equal to f(r).
So I attempted to find r by factorizing x2−x+1 first but I got the complex form x=1±√3i2=r.
f(r) is then (1+√3i2)3 or (1−√3i2)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)(x2−x+1) so x3=(x+1)(x2−x+1)−1 the answer is −1.
Similarly for x3n+1=(x3+1)((x3)n−1−(x3)n−2+...−(x3)+1)⏟q(x)=(x+1)(x2−x+1)q(x)
so the answer is again −1.
No comments:
Post a Comment