Question

What are conditions for cubic polynomial have 3 real roots?

Answer 1

Answer:

a + b + c = - b/a

ab + bc + ca = c/a

abc = - d/a

Answer 2

For a Cubic Equation :

$f(x) = a {x}^{3} + b {x}^{2} + cx + d = 0$

$\forall \: \: x_{1,2,3} \in \: ℝ$

The Condition For All Real Roots to Occur :

$\sf{b}^{2} {c}^{2} + 18abcd - 27 {a}^{2} {d}^{2} - 4a {c}^{3} - 4 {b}^{3} d > 0$

⇒ All the Roots of the Cubic Are :

$\sf \: x_{1,2,3} = \frac{ - b \: + \:2 \sqrt{ {b}^{2} - 3ac } \cos( \frac{1}{3} \arccos [ \frac{9abc - 2 {b}^{3} - 27 {a}^{2} d}{2( {b}^{2} - 3ac) \sqrt{ { {b}^{2} - 3ac} } } ] + \frac{2k\pi}{3} ) }{3a}$

where,

$k \in (0,1,2)$

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