Question

Look at the following polynomials and their factorizations:

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)

In general, how can x^n-1 be factored. Show that this factorization works by multiplying the factors

Answer 1

Just do a long multiplication and line up the like exponents:

You'll notice that all the exponents have a plus entry and a minus entry, except the highest and lowest:

x^n + x^n-1 + x^n-2 + ... + x^2 + x^1
000 - x^n-1 - x^n-2 - ... - x^2 - x^1 - 1
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x^n - 1