- #1

SwimmingGoat

- 7

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[itex]q(x)=x^2-14\sqrt{2}x+87[/itex]. Find 4th degree polynomial [itex]p(x)[/itex] with

__integer__coefficients whose roots include the roots of [itex]q(x)[/itex]. What are the other two roots of [itex]p(x)[/itex]?

I found that the two roots of [itex]q(x)[/itex] are [itex]x=7\sqrt{2}-\sqrt{11}[/itex] and [itex]x=7\sqrt{2}+\sqrt{11}[/itex]. Since they are conjugates of each other, I have no idea what to guess the other roots could be of my fourth-degree polynomial.

I started out with trying to get rid of the [itex]14\sqrt{2}[/itex] like so:

[tex](x^2-14\sqrt{2}+87)(x+14\sqrt{2})[/tex] but I ended up with

[tex](x^3+87x+1218\sqrt{2}-392)[/tex] Going to the fourth degree looked like a headache, and I felt I wasn't on the right track, so I stopped there.

Any ideas?