As for the last one... Look at the polynomial . You can see that the sum of the coefficients is 0. So 1 is a root of it.
Hence you can factor by
Actually, after some calculations,
And you can do partial fractions here
This one gets a little bit tricky. So first off I do long division to make the fraction proper:
Here is my result. You can check my division although i'm not skilled enough in MathType to show all of it yet.
Now I need to find out a way to simplify that last term.
This is as far as I went. Any help is appreciated. Thanks!
If the sum of the coefficients is 0, you can say that 1 is a root, that is to say :
More generally, let a polynomial , with the sum of coefficients equal to 0.
If we let x=1, we'll have and this equals 0 since the sum of all coefficients is 0. hence 1 is a root.
Now, if a is a root of a polynomial Q(x), then Q(x) is a multiple of (x-a).
It's a bit messy, but it should be all I can say on the subject :s