Is it possible to factor 1+x^5?

The answer to both questions is yes, it is possible.

However, unless I am missing something, actually integrating this expression is an extremely tedious, difficult calculation.

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For your first question: Yes, you can factor it, but it requires some advanced techniques for factoring.

First off, you can factor 1 + x^5 using the technique for a sum of two quantities raised to the same odd power (see http://en.wikipedia.org/wiki/Factorization ):

1 + x^5 = (1 + x)(x^4 - x^3 + x^2 - x + 1).

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It turns out that you can further factor x^4 - x^3 + x^2 - x + 1:

(1 + x)(x^4 - x^3 + x^2 - x + 1) = (1 + x)(x^2 + ((-1 + √5) / 2)x + 1)(x^2 + ((-1 - √5) / 2)x + 1)

(I came upon this factorization just by guessing a factorization for x^4 - x^3 + x^2 - x + 1 of the form (x^2 + Cx + 1)(x^2 + Dx + 1), and then multiplying this out and solving for what C and D had to be.)

No more factorization is possible over the real numbers, because in each of the quadratic factors, the discriminant is negative (that is, the quadratic formula yields a negative square root).

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If you want to continue factoring over the complex numbers using the quadratic formula on the two remaining quadratic factors, you can, but this gets pretty ugly (square roots inside square roots), so I'll just content myself with saying it's possible.

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To answer your second question: It is possible to integrate x^2 / (1 + x^5) by using partial fractions. First, factor the denominator:

x^2 / (1 + x^5) = x^2 / [(1 + x)(x^2 + ((-1 + √5))/2)x + 1)(x^2 + ((-1 - √5)/2)x + 1)]

Then use partial fractions to decompose the fraction:

x^2 / [(1 + x)(x^2 + ((-1 + √5))/2)x + 1)(x^2 + ((-1 - √5)/2)x + 1)]

= A / (1 + x) + (Bx + C) / (x^2 + ((-1 + √5)/2)x + 1) + (Dx + E) / (x^2 + ((-1 - √5)/2)x + 1)

...and you can solve for A, B, C, D, and E in the usual way that you would in partial fractions. This would involve solving a system of five linear equations in five variables--tedious but possible.

When you get finished, you can integrate each piece by using various substitutions. Again, this computation would be very tedious and difficult, but in principle it is possible.

Computations this tedious are usually best left to a computer, in my opinion. The final result of the integral is given here:

http://integrals.wolfram.com/index.jsp?expr=x%5E2+...

It's really impossible to factor 1+x^5

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RE:

Is it possible to factor 1+x^5?

Or to integrate

(x^2)/(1+x^5)?

Thanks!