Complex solution of a polynomial

[Guest]

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]

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 [tex]a+bi[/tex] is a zero, then so is the conjugate [tex]a-bi[/tex] ...

[tex][z-(3+i)][z-(3-i)] = z^2 - 6z + 10[/tex]

[HallsofIvy]

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 [tex]z^2- 6z+ 10[/tex] as a factor. Since the given polynomial is of 4th degree the other factor must also be a quadratic polynomial so or the form [tex]z^2+ pz+ q[/tex] for some numbers p and q. That is, we must have [tex](z^2- 6z+ 10)(z^2+ pz+ q)= z^4- 2z^3+ az^2+ bz+ 10[/tex].

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 [tex]z^2+ pz+ q= 0[/tex].