Question

What is the equation of a quadratic function p with rational coefficients that has a zero of 3-i

Answer 1

The equation of the quadratic function p with rational coefficients that has a zero of 3-i is p(x) = (x - 3+i)(x - 3-i).

Step-by-step explanation:

The equation of the quadratic function p with rational coefficients that has a zero of 3-i can be found using the concept of complex conjugate roots.

Since the coefficients of the quadratic function are rational, the other root must be the complex conjugate of 3-i, which is 3+i.

Using these two roots, the equation of the quadratic function can be written as: p(x) = (x - 3+i)(x - 3-i).

Answer 2

p = x² − 6x + 10

Step-by-step explanation:

Complex roots come in conjugate pairs. So if 3−i is a root, then 3+i is also a root.

p = (x − (3−i)) (x − (3+i))

p = x² − (3+i)x − (3−i)x + (3−i)(3+i)

p = x² − 3x − ix − 3x + ix + (9 − i²)

p = x² − 6x + 10

You can check your answer using the quadratic formula.

x = [ -b ± √(b² − 4ac) ] / 2a

x = [ 6 ± √(36 − 40) ] / 2

x = (6 ± 2i) / 2

x = 3 ± i