Question

Knowing that the remainder of the division of x³ by x²-x+1 is -1, the remainder of x^2007 by x²-x+1 is...?

Please, I'd really appreciate an explanation , thank you!

Answer 1

either 1 or -1 since long division has a cycle of 4 in order to return back to a positive exponent
since 2007 -1 = 2006 and the closet factor of 4 is 2004 leaving 2

it would end in a negative

i think?

Answer 2

since 2 means it ends in a half cycle and a half cycle results in a negative value

Answer 3

You can apply the algorithm performed here to discern a pattern: https://en.wikipedia.org/wiki/Synthetic_division#Expanded_synthetic_division
\[\small\begin{array}{cc|ccccccccccc}
\text{deg}&&2007&&&2004&&&2001&&&1998\\[1ex]
\hline
&&1&0&0&0&0&0&0&0&0&0&0&0&\cdots\\[1ex]
&-1&\color{lightgray}{\downarrow}&&\color{red}{-1}&\color{orange}{-1}&\color{yellow}{0}&\color{green}{1}&\color{blue}{1}&\color{indigo}{0}&\color{violet}{-1}&\color{red}{-1}&\color{orange}{0}&\color{yellow}{1}&\cdots\\[1ex]
1&&\color{lightgray}{\downarrow}&\color{red}{1}&\color{orange}{1}&\color{yellow}{0}&\color{green}{-1}&\color{blue}{-1}&\color{indigo}{0}&\color{violet}{1}&\color{red}{1}&\color{orange}{0}&\color{yellow}{-1}&\color{green}{-1}&\cdots\\[1ex]
\hline
&&1&1&0&-1&-1&0&1&1&0&-1&-1&0&\cdots
\end{array}\]Notice that for every column where the \(n\)th degree is an odd multiple of \(3\), the coefficient of the \((n-1)\)th power term in the quotient is positive. This means the right end of the table will look like this:
\[\small\begin{array}{cc|ccccccc|cc}
\text{deg}&&2007&&&2004&\cdots&2&1\\[1ex]
\hline
&&1&0&0&0&\cdots&0&0&0&0\\[1ex]
&-1&&&-1&-1&\cdots&1&0&-1&-1\\[1ex]
1&&&1&1&0&\cdots&0&1&1\\[1ex]
\hline
&&1&1&0&-1&\cdots&1&1&0&-1
\end{array}\]so the remainder will still be \(-1\).

I'm sure there's any easier, not-so-long-winded way to arrive at this result, but this works.

Answer 4

How did you make those perfect pictures HolsterEmission?

Answer 5

Lots of practice in using TeX :)

If you're curious about the code, you can right click it, pick "Show Math As..." and "TeX Commands".

Answer 6

@sooobored How do you know that this division has a cycle of 4 in order to return back to a positive exponent?

Answer 7

intuition isnt a good answer

mathematical inference, (and ive done some of these types of problems when i was younger)

i noticed that a single division will result in a polynomial (more than one term)
a second division will result in a monomial but of the opposite sign and that monomial is closely related to the original monomial we started with

so I make the assumption, for every 2 division, we reduce the exponent by 3 and change the sign

which means if we want to go from a negative to a positive, we would require an additional 2 more divisions

thus to go from a positive to a positive through division, we would need a cycle of 4 division, and every 4 cycles would reduce the exponent by 6

i realize, i mightve made a mistake in the above calculations, but this is the overall thought process

Answer 8

The other way to determine a cycle of 4 would be to actually do 4 consecutive divisions to determine whether you would return to a positive

Answer 9

@sooobored so, I did 4 consecutive divisions and it returned a remainder of \[x ^{n-6}\] , but how can I guarantee that the pattern will always repeat at the 4th cycle ?

Answer 10

sorry, the term was induction

basically, if you see a pattern, you make a guess that the pattern is recurring and you do your best to try to prove it wrong, but if you cant, you assume the pattern is ongoing

Answer 11

so basically the thought process up to the 4th division should be

wow, i notice that every 2 division reduces the exponent by 3 and changes the sign

i noticed that an additional 2 division further reduces the exponent by 3 and changes the sign, resulting in every 4 division reduces the exponent by 6 and the sign doesnt change

then the question is, will this pattern continue for every 2 division

and you do a couple more division until you're absolutely sure this pattern doesnt change

there's no way to guarantee it wont follow the pattern, but you also have to ask yourself if im changing what im dividing by, will i still get the same pattern

if i dont change what im dividng by and there is no additional steps im adding, then why would this pattern change?

Answer 12

@sooobored and @HolsterEmission I understand it now, thanks for your help!