A Real Equation!

Algebra Level 4

If f(x)=a1x3+a2x2+a3=0 where a1,a2R+ and a3R\displaystyle f(x) = a_1x^3 + a_2x^2 + a_3 = 0 \text{ where } a_1, a_2 \in \R^+ \text{ and } a_3 \in \R has three distinct real roots, then the exhaustive range of a3\displaystyle a_3 is a3(ba2cda1e,f) where b,c,d,e,fI and gcd(b,d)=1\displaystyle a_3 \in \bigg(-\frac{ba_2^c}{da_1^e}, -f\bigg) \text{ where } b, c, d, e, f \in I \text{ and } \gcd( b, d) = 1, enter answer as b+c+d+e+fb + c + d + e + f.


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