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# 57. Divide. Look for patterns in your answers. Can someone explain what are these patterns?

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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.

Feb 18, 2019

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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)   Feb 18, 2019