The document provides steps for dividing x101 + x + 1 by x2 + 1 by looking for a pattern in the exponents. - The exponents of the answer decrease by two and switch signs with each term, following the pattern of X99 -x97 + x95-x93... - Bringing the pattern all the way down allows the remainder to be determined as 2x + 1 when the division is completed.

1. Kenny’s POTW Solution
3/29/10

2. • When dividing x101 + x+ 1 by x2 + 1, look for a
pattern.
X99 -x97 + x95-x93…
x2+ 1 x101+ 0x100+0x99 … + x+1
-x101 –x100
-x99 + 0x99
+x99 + x97
+x97 +0x98
-x97 - x95
-x95…

3. • The exponents of the answer are decreasing
by two and switching signs every term.
• If you bring the pattern all the way down to
terms that can work with the x + 1 in the
dividend, you end up with… x3 –x.
• Taking this end of the answer, you can apply it
to the end of the dividend. By using the
opposite of x3 –x, you work backwards in the
division like so…

4. … x3 –x +1 •Since the patter was
x2+ 1 …x5 –x3 +x +1 also followed under
-x5 – x3 the dividend, it can be
concluded that –x3 +x
-x3 +x +1 will follow. When
+x3 +x the division is
2x +1 completed, the
remainder becomes
2x+1/ x2+ 1