Question

Find a third-degree polynomial equation with rational coefficients that has roots -4 and 6+i

Answer 1

If one root is 6+i, to keep rational coefficients you have to make the other multiply with it to produce a rational number. That means you need the complex conjugate (same real component, negative imaginary component)

Answer 2

So 6-i would be the complex conjugate?

Answer 3

Right. Then, set the polynomial as equal to x- (each root) multiplied with each other.

Answer 4

(x-4)(x-6-i)(x-6=i) is that right?

Answer 5

I meant (x-6+i) for the last one

Answer 6

Yeah. Then multiply it out (I'd start with the terms that include i) and you've got your polynomial. You can multiply the coefficients by any scalar afterwards.

Answer 7

So i'd start off by multiplying (x-6-i)(x-6+i)

Answer 8

I'm not sure how to multiply that.

Answer 9

You know FOIL? Basically make sure each term multiplies every other term.