SOLUTION: Show that (x-1) (x+1) is a factor of f(x)=3x^50-3

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Question 1177057: Show that (x-1) (x+1) is a factor of f(x)=3x^50-3
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29=3x%5E50-3
f%28x%29=3%28x%5E50-1%29
zeros are
3%28x%5E50-1%29=0....since factor 3%3C%3E0, we have
x%5E50-1=0
x%5E50=1
x=root%2850%2C1%29.....no matter what root is it, solutions are
x=1 or x=-1
using root product theorem, we have
%28x-1%29+%28x%2B1%29
so, it is a factor of f%28x%29=3x%5E50-3

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

3x^50 - 3 = 3*(x^50-1) = 3*((x^2)^25-1) = 3*(x^2-1)*(1 + x^2 + x^4 + x^6 + . . . + x^48).


Here I used the formula for the sum of an geometric progression


    1 + z + z^2 + z^3 + . . . + z^(n-1) = %28z%5En-1%29%2F%28z-1%29


with z = x^2.

Solved.