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A complex root of a polynomial can have some significance itself when the roots of the polynomial have significance in general However, even with the rational roots test and sythentic division, the guess part of the process is a little unappealing to me. One example that comes to mind where the roots of polynomials have a meaningful interpretation is in the field of dynamical systems.
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The axis of symmetry is the real part of the complex roots Apparently, one valid method is to try to guess one of the roots and then use it to divide the polynomial This assumes the roots come in conjugate pairs (so the coefficients of your quadratic are real numbers).
We can present complex roots to equation on the complex plane with one axis for the real part and the other for the imaginary part
You can play with, for instance, wolframalpha, to give it a polynomial equation to solve and get a display of the complex roots. Applied to quartic equations with two sets of complex conjugate roots, the theorem implies that in general the roots of the quartic are at the vertices of a quadrilateral in the complex plane and the roots of the derivative (real and otherwise) lie inside this quadrilateral. I think i will start by demonstrating the distributive and commutative properties of the complex conjugate, then using those to state that if a complex numbers is a root of a polynomial, so is its conjugate, and therefore complex roots must come in conjugate pairs Then i can just list the cases for each order polynomial.
The square root of i is (1 + i)/sqrt (2) [try it out my multiplying it by itself.] it has no special notation beyond other complex numbers How to do partial fraction decomposition with complex roots Ask question asked 6 years, 9 months ago modified 6 years, 9 months ago
0 when i tried to solve the given cubic equation i found that it has three real roots as hereunder
Since you have mentioned that the equation has complex roots i tried to put the above values in the equation and found that the equation is satisfied with each value.
