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

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Expert Answers

cburr | Certified Educator

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**.