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# 56. Reasoning Divide. Look for patterns in your answers. Can someone expalin what are these patterns?

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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.

Feb 18, 2019

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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

_______

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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.....!!!!   Feb 18, 2019