Cardans_Method - Extracting Roots from Cubic Polynomials Cardans Method Consider any cubic polynomial in x a3x3 a2x2 a1x ao = 0 This polynomial has

Cardans_Method - Extracting Roots from Cubic Polynomials...

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Extracting Roots from Cubic Polynomials - Cardan’s Method Consider any cubic polynomial in x , a 3 x 3 + a 2 x 2 + a 1 x + a o = 0 This polynomial has three roots; one must be real and the other two may be real or imaginary. The three roots can be found analytically using Cardan’s Method, which is described below. To obtain the real roots to any cubic having real coefficients, first write your cubic in this form: x 3 + bx 2 + cx + d = 0 Then compute p = 2 b 3 " 9 bc
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Unformatted text preview: 9 bc + 27 d 54 and q = b 2 " 3 c 9 If ( p 2 – q 3 ) > 0, then the cubic has only one real root, x = " sgn( p ) r 2 + q r # $ % & ’ ( " b 3 where sgn( p ) = p /| p | and r = p 2 " q 3 + p ( ) 1/ 3 But if ( p 2 – q 3 ) ≤ 0, then the cubic has three real roots, x 1 = (-2 q )cos( θ /3) – ( b /3) x 2 = (-2 q )cos[( θ + 2 π )/3] – ( b /3) x 3 = (-2 q )cos[( θ + 4 π )/3] – ( b /3) where " = cos # 1 p q 3 ( )...
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