Find a third-degree polynomial function with real coefficients and with zeros 1 and 3+i

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To write a 3rd degree polynomial, you can write it in terms of its roots:

$$f(x) = (x-r_1)(x-r_2)(x-r_3)$$

You can see if you plug in any of the roots for x, it will make the whole thing zero, which is what a root does so for example plug in $x=r_2$ and you get  $f(r_2) = 0$ Try it out to make sure.

Ok, so let's create our polynomial now, plug in the roots.

You'll see that you were only given two roots, so what's the 3rd one?  Well in order to make a real polynomial you have to have the 3rd root be the complex conjugate of the one imaginary root, since that will make sure we have a real number.