| ElizS66 |
How do you know how many roots a cubic equation has? e.g. How many solutions does the equation x^3 + ax^2 - x - 2 = 0 have, for x > 0, or does this depend on a? Do you differentiate it? ElizS66 |
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| Chris
Tynan |
A cubic equation can always be factored into the form (x-a)(x-b)(x-c) by the fundamental theorem of algebra, even though some of a,b,c may be complex numbers. However, you're interested in whether certain solutions are the same. Say if two solutions are the same, then we have (x-a)^2(x-b) = x^3-(2a+b)x^2+(a^2+2ab)x-a^2b. So if your equation is of this form, where a=/= b, then there are two unique solutions. Likewise, if your equation is of the form x^3-3ax^2+3a^2x-a^3 then there is just one unique solution. Otherwise, unless I'm mistaken, there are always three unique solutions. |
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| ElizS66 |
Thanks - that really helped. Also, how do you know if the roots are real or not? ElizS66 |
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| Kerwin
Hui |
The discriminant of a cubic (x-a)(x-b)(x-g) is defined to be D=(a-b)2(a-g)2(b-g)2 (the discriminant is defined for all polynomials of degree at least 2, as the product of all squares of possible differences of roots). By some algebraic manipulation, the discriminant of the cubic x3 + A x2 + B x + C is D=A2 B2 - 4C A3 - 4B3 + 18A B C - 27 C2 If all coefficients are real, then: D=0 if we have repeated roots D > 0 if we have 3 real roots D < 0 if we have 1 real root and 2 nonreal roots. Kerwin |