SOLUTION: The polynomial p(z)=2zł+az˛+bz-5 where a and b are real has 2-i as one of its roots. find the value of the constants a and b. If you could show me how to solve this, thatd be great

Algebra ->  Algebra  -> Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: The polynomial p(z)=2zł+az˛+bz-5 where a and b are real has 2-i as one of its roots. find the value of the constants a and b. If you could show me how to solve this, thatd be great      Log On

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Question 593317: The polynomial p(z)=2zł+az˛+bz-5 where a and b are real has 2-i as one of its roots. find the value of the constants a and b. If you could show me how to solve this, thatd be great, cheers.
Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
complex roots (containing i) occur in conjugate pairs ___ if 2-i is a root, 2+i is also

if r is a root, then z-r is a factor

the product of factors is a factor ___ (z - 2 + i)(z - 2 - i) = z^2 - 4z + 5

to get a coefficient of 2 for z^3 and a -5 for the constant term, the 3rd factor must be 2z-1

(2z - 1)(z^2 - 4z + 5) = 2z^3 - 8z^2 + 10z - z^2 + 4z - 5 = 2z^3 - 9z^2 + 14z - 5