SOLUTION: One root of the equation z3 – 10z2 + 37z + p = 0, where p is real, is z = 3 + 2i. Find the value of p and the other two roots.

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: One root of the equation z3 – 10z2 + 37z + p = 0, where p is real, is z = 3 + 2i. Find the value of p and the other two roots.      Log On


   

Question 242489: One root of the equation z3 – 10z2 + 37z + p = 0, where p is real, is z = 3 + 2i.
Find the value of p and the other two roots.
Answer by solver91311(24713) About Me  (Show Source):
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where and one root is

If is a root, then is also a root because complex roots always appear in conjugate pairs,

If is a root of a polynomial equation, then is a factor of the polynomial. So:

Multiply the two known factors of the polynomial:



Verification of this last step is left as an exercise for the student.

The product of the two known factors can be divided into the original polynomial equation using polynomial long division. If you are unfamiliar with polynomial long division or you need a refresher on the subject see:

http://www.purplemath.com/modules/polydiv2.htm

or

http://en.wikipedia.org/wiki/Polynomial_long_division

As you complete the division process, choose so that there is no remainder from the division. If there is no remainder, then the quotient is the remaining factor from which the third root can be determined directly.

John