Complex solution of a polynomial

Complex solution of a polynomial

Hey guys, here is my problem and I'm not sure how to solve it

3+i is a solution of z^4-2z^3+az^2+bz+10 = 0. How do I solve for a and b and find the remaining solutions over real and complex.
Guest

Re: Complex solution of a polynomial

Guest wrote:Hey guys, here is my problem and I'm not sure how to solve it

3+i is a solution of z^4-2z^3+az^2+bz+10 = 0. How do I solve for a and b and find the remaining solutions over real and complex.

If $$a+bi$$ is a zero, then so is the conjugate $$a-bi$$ ...

$$[z-(3+i)][z-(3-i)] = z^2 - 6z + 10$$
Guest

Re: Complex solution of a polynomial

To continue what "guest" wrote (he sure does post a lot here!) any polynomial, with real coefficients, that has 3+ i as roots must also have 3-i as a root and so must have $$z^2- 6z+ 10$$ as a factor. Since the given polynomial is of 4th degree the other factor must also be a quadratic polynomial so or the form $$z^2+ pz+ q$$ for some numbers p and q. That is, we must have $$(z^2- 6z+ 10)(z^2+ pz+ q)= z^4- 2z^3+ az^2+ bz+ 10$$.

Do the indicated multiplication on the left and compare coefficients to determine what p, q, a, and b must be and solve the quadratic equation $$z^2+ pz+ q= 0$$.

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