+73 56. Reasoning Divide. Look for patterns in your answers. Can someone explain what are these patterns?
a. (x^2 - 1) / (x - 1)
b. (x^3 - 1) / (x - 1)
c. (x^4 - 1) / (x -1)
d. Using the patterns, factor x^5 - 1.
+129852 OK.....we should be able to see the pattern fairly quickly
Note that x^2 - 1 factors as (x - 1) ( x + 1)
So
(x^2 - 1) / (x - 1) =
(x+ 1) ( x - 1) /(x - 1) = x + 1
And
x^2 + x + 1
x - 1 [ x^3 + 0x^2 + 0x - 1 ]
x^3 - x^2
_________________
x^2 + 0x
x^2 - 1x
_____________
1x - 1
1x - 1
_______
0
And
x^3 + x^2 + x + 1
x - 1 [ x^4 + 0x^3 + 0x^2 + 0 x - 1 ]
x^4 - 1x^3
__________________________________
1x^3 + 0x^2
1x^3 - 1x^2
_________________
1x^2 + 0x
1x^2 - 1x
___________
1x - 1
1x - 1
_______
0
Notice the pattern, GM
(x^2 -1) / (x - 1) = x + 1
(x^3 - 1) / (x - 1) = x^2 + x + 1
(x^4 - 1) / (x -1) = x^3 + x^2 + x + 1
So......this seems to imply that
(x^5 - 1) / (x - 1) = x^4 + x^3 + x^2 + x + 1
Check for yourself that this is true.....!!!!
