Solving cubic equation with one variable Find the solution of MX ^ 3 + NX ^ 2 + LX + o = 0, the equation has only one real root, between 0 and 1 M. N, l, O are integers, and the sum of them is a fixed value, and the coefficients of the quadratic equation of one variable formed by their complex solutions are integers. The further relationship between the fixed value and m, N, l, O is similar to the relationship between a, B, C in the expression of Pythagorean theorem Let me change my question. A cubic equation with one variable must have a unique real root. MX ^ 3 + NX ^ 2 + LX + O has factorization (AX + b) (Cx ^ 2 + DX + e). E is an integer, and the fixed value is a prime number. There are only two prime factors. To find the formula like Pythagorean theorem, we can find the expression that m and O can be expressed by several other independent factors , cancel the real root between 0 and 1

Solving cubic equation with one variable Find the solution of MX ^ 3 + NX ^ 2 + LX + o = 0, the equation has only one real root, between 0 and 1 M. N, l, O are integers, and the sum of them is a fixed value, and the coefficients of the quadratic equation of one variable formed by their complex solutions are integers. The further relationship between the fixed value and m, N, l, O is similar to the relationship between a, B, C in the expression of Pythagorean theorem Let me change my question. A cubic equation with one variable must have a unique real root. MX ^ 3 + NX ^ 2 + LX + O has factorization (AX + b) (Cx ^ 2 + DX + e). E is an integer, and the fixed value is a prime number. There are only two prime factors. To find the formula like Pythagorean theorem, we can find the expression that m and O can be expressed by several other independent factors , cancel the real root between 0 and 1

Don't use factorization, because it can only deduce the relationship between ABCDE and mnlo. It's too complicated to deduce the relationship between mnlo from this relationship
First, use the discriminant of univariate cubic equation for analysis (but it is also very complex)
The univariate cubic equation is m x ^ 3 + n x ^ 2 + L x + o = 0
Shilling
A = N^2 - 3*M*L (1)
B = N*L - 9*M*O (2)
C = L^2 - 3*N*O (3)
Discriminant: delta = B ^ 2 - 4 * a * C (4)
When delta > 0, there is only one real root and two imaginary roots
x1 = ( - N - Z )/(3M) (5)
x2 = ( -4N + 2*M*Z + i * Z/sqrt(3) ) / (2M) (6)
x3 = ( -4N + 2*M*Z - i * Z/sqrt(3) ) / (2M) (7)
among
Z = Y1^(1/3) + Y2^(1/3) (8)
among
Y1 = A*N + 3*M/2 * ( -B + sqrt(delta) ) (9)
Y2 = A*N + 3*M/2 * ( -B - sqrt(delta) ) (10)
To meet the requirements of the topic
X1 between 0 and 1: 0 < (- N - z) / (3m)