+73 57. Divide. Look for patterns in your answers. Can someone explain what are these patterns?
a. (x^3 + 1) / (x + 1)
b. (x^5 + 1) / (x + 1)
c. (x^7 + 1) / (x + 1)
d. Using the patterns, factor x^9 + 1.
+128473 x^2 - x + 1
x + 1 [ x^3 + 0x^2 + 0x + 1 ]
x^3 + 1x^2
_____________________
-1x^2 + 0x
-1x^2 - 1x
____________
1x + 1
1x + 1
_________
0
x^4 - x^3 + x^2 - x + 1
x + 1 [ x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 1 ]
x^5 + 1x^4
_______________
-1x^4 + 0x^3
-1x^4 - 1x^3
__________________
1x ^3 + 0x^2
1x^3 + 1x^2
_______________
-1x^2 + 0x
-1x^2 - 1x
_____________
1x + 1
1x + 1
_______
0
Note GM that the pattern seems to be one of alternating signs on decreasing powers
(x^n + 1) / ( x + 1) = x^(n - 1) - x^(n - 2) + x^(n - 3) - x^(n - 4) + ..... - x .+ 1
So.....we can intuit that
(x^7 + 1) / ( x + 1) = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Knowing this.....can you find (x^9 + 1) / ( x + 1) ?????
And finding that.....the factorization will be ... (your answer ) (x + 1)
