Question

In the equation x^3 + rx^2 + sx + t =0 where r,s,and t are integers and 3i and -3i are roots, tell whether the statement is never, always, or sometimes true.

Answer 1

If 3i and -3i are roots then (x-3i)(x+3i)=x^2+9 is a factor. The remaining factor is x+a, say, so (x^2+9)*(x+a)=x^3+9x+ax^2+9a, and this is equal to x^3+rx^2+sx+t. So a=r and s=9, and t=9a=9r and therefore r and t are related and s=9. You could also say t=rs, so all three are related, although s must equal 9. The statement is true sometimes, and only when the conditions s=9 and t=rs apply.