(x^n - 1)=(x-1)(x^n + x^(n-1) + ... + x + 1)?

Question

Uuse distributivity on the right-hand side of the equation, until all of the parentheses are gone. I'm thoroughly confused.

Answer 1

Obviously, the term on the left side of the equals sign is okay as it it.

On the right, first get it clear in your head that the (x - 1) is your first term.  The whole thing!    The second term is the (x^n + x^(n - 1) + ... + x + 1).  The whole thing!  This is often where people go wrong.  Also, since you are working with exponents it is important to remember that x by itself is an implied x^1.

The first thing to do is multiply the x in the first term by each of the components of the second term.  Like this...

x times x^n gives you 2x^n  and then x times x^(n - 1) gives you 2x^(n - 1) and so forth.  (NOTE FOR FUTURE PROBLEMS:  Once you have multiplied the x times each of the parts of the second term, don't go further!  Don't make a mistake and multiply the x by any of the terms OUTSIDE the second term.)  Then do the same thing with the -1 and so on in the first term.  Then gather like terms together, if any, and that is your answer.

(x - 1) is the first term, and (x^n + x^(n-1) + ... + x + 1) is the second term.

I have found an excellent site for helping to explain problems like these and it also has a calculator that will figure out problems for you and lay out all the steps so you can easily follow what was done to reach the answer.  Click the link below and read all the pages of the lesson so you will thoroughly understand.  Then click on the calculator tab at the top.

Good luck!