find a fourth degree polunomial equation,?

Question

find a fourth degree polynomial equation with integer coefficients, which has no cubic term and no constant given: 3-i√7 is one root.

show all work!!!

Answer 1

If 3-i√7is one root, then 3+i√7 must be another.
Hence (x^2 -6x+16) must be a factor of the quartic equation.
Let the other factor be ax^2+bx+c
Then the quartic is:
ax^4 -(6a+b)x^3 +(16a+6b+c)x^2 -(16b+6c)x + 16c=0
Since there is no constant term, c = 0 and equation becomes:
ax^4 -(6a+b)x^3 +(16a+6b)x^2 -16bx = 0
Since there can be no x^3 term b=-6a so equation becomes
ax^4 +(16a-36a)x^2 + 96x =0
Let a =1, then equation is:
x^4 -20x^2+96x = 0

Answer 2

7x^4+5x^2+6x