2. “How many different polynomials of the form x 4 + ax 3 + bx 2 + cx + d
With rational coefficients are factors of the polynomial x -1
12 “
First, factor x12 -1 into ( x -1)( x +1)( x 2 +1)( x 2 - x +1)( x 2 + x +1)( x 4 - x 2 +1)
f1 f2 f3 f4 f5 f6
To make notation easier, we will call each factor f1 through f6 in the order given above
Next, multiply the factors together to make polynomials of the form x 4 + ax 3 + bx 2 + cx + d
There are 7 of these polynomials that can be formed:
1.) f1*f2*f3 = x 4 -1
2.) f6 = x 4 - x 2 +1
3.) f1*f2*f4 = x - x + x -1
4 3
4.) f1*f2*f5 = x + x - x -1
4 3
5.) f3*f4 = x - x + 2x - x +1
4 3 2
6.) f4*f5 = x 4 + x 2 +1
7.) f3*f5 = x + x + 2x + x +1
4 3 2
