Factor the 5 term polynomial X^101+X^52+X^50+X^2-1

One method is guess-and-check. The given polynomial has five terms, so binomial factors won't work (two binomial factors yield four terms, at most).

We could guess trinomial factors.

X^101 + X^52 + X^50 + X^2 - 1 = (? + ? + ?) (? + ? + ?)

The product of the red terms must be -1, yes?

X^101 + X^52 + X^50 + X^2 - 1 = (? + ? + 1) (? + ? - 1)

The product of the blue terms must be X^2. (Can you see why?) Therefore, you know what they are, yes?

The product of the black terms must be X^101. Now it's time to guess and check …
 
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