SOLUTION: I am not sure how to factor this problem: X^4-X^3+X-1

Question 401037: I am not sure how to factor this problem:
X^4-X^3+X-1
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

There is no Greatest Common Factor (GCF) other than 1. And there are too many terms for factoring by patterns or for trinomial factoring. All that is left is factoring by grouping or by trial and error of the possible rational roots. The trial and error method is usually a method of last resort. So we will try factoring by grouping.

When factoring by grouping, I find it helps to rewrite any subtractions as additions of the opposite. By having an expression of all additions we can use the Commutative and/or Associative Properties as needed and this method of factoring sometimes needs rearrangement of the terms. Changing the subtractions sometimes helps in another way as we will see shortly. So first we make everything an addition:

Now we can group the terms. We want pairs of terms from which we can factor out a common factor. (Note: Factoring by grouping is one of the few occasions where you might actually factor out a 1!)

From the first group we can factor out . From the second group we can factor out a 1 or a -1. I'll factor out a 1 because I can see that it is the one we need:

As you can see, the "other" factors in each group are the same. This is what we want when factoring by grouping. We can now factor out (x+(-1)):

or
}

When factoring, keep factoring until no more factoring can be done. The first factor above will not factor further. But the second factor is a sum of cubes and will factor according to the pattern :

which simplifies to:

No more factoring can be done so we are finished.
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