If the distance from any point P to each side of positive n-sided a1a2a3. An is R1, R2, R3... RN, is R1 + R2R3 +... RN a fixed value? If so, please guess the fixed value reasonably

If the distance from any point P to each side of positive n-sided a1a2a3. An is R1, R2, R3... RN, is R1 + R2R3 +... RN a fixed value? If so, please guess the fixed value reasonably

It is the fixed value
Consider that the area of a positive n-edge is equal to n * (each side multiplied by its height divided by 2), and because each side is the same, it is equal to (the sum of the distances from the side length * P to each side) * n / 2,
Since the area is fixed as s and the side length is r, then the fixed value is (s * 2) / (n * r)

It is proved that the ideal (2, x) of polynomial ring Q [x] over rational number field q is the principal ideal

Modern algebra is a headache. There are some problems on douding.com

Can this polynomial be decomposed in rational number field?
f(x)=x^4-5x+1
I've tried Eisenstein's method and several variants, but I still can't tell

If we can prove that x ^ 4 + X + 1 is irreducible in F 2, then we can prove that x ^ 4 + X + 1 is irreducible in F 2
First of all, x ^ 4 + X + 1 is my root in F2, so if it can be decomposed, it must be the product of irreducible polynomials of degree 2
In F2, the irreducible polynomial of degree 2 has only x ^ 2 + X + 1, but (x ^ 2 + X + 1) ^ 2 = x ^ 4 + x2 + 1 is not equal to
X ^ 4 + X + 1 is proved