Which is the term of the expanasion (x-1)^13, which doesn't have the unknown x?

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cburr | Middle School Teacher | (Level 2) Associate Educator

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The full expansion of (x-1)^13 is:

x^13  -  13x^12  +  78x^11  -  286x^10  +  715x^9  - 1287x^8  +  1716x^7  - 1716x^6 + 1287x^5  -  715x^4  +  286x^3  -  78x^2  +
13x  -   1

It's not really necessary to do the full expansion to know that the x-free term will be either + or - 1 -- there is no other number possible when all you are doing is multiplying -1 times itself repeatedly.

To figure out the + or -, think about the pattern.

-1 x -1 = 1      one time

1 x -1 = -1     two times

-1 x -1 = 1      three times

etc.

Since there are 13 (x-1)s in a row, you are multiplying by -1 a total of 12 times (the first (x-1) is the starting point).  As you can see above, when you multiply an even number of times you end up with -1.

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