100 POINT QUESTION!!! Please help!
2 answers:
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It looks like you might have intended to say the roots are 7 + i and 5 - i, judging by the extra space between 7 and i.
The simplest polynomial with these characteristics would be
but seeing as each of the options appears to be a quartic polynomial, I suspect f(x) is also supposed to have only real coefficients. In that case, we need to pair up any complex root with its conjugate to "complete" f(x). We end up with
which appears to most closely resemble the third option. Upon expanding, we see f(x) does indeed have real coefficients:
Uhh, answer choices?
In any case, since is inscribed in a semicircle, it is a right triangle (due to the inscribed angle theorem). Thus, by the Pythagorean theorem,
Now, the circumradius of a right triangle (the radius of the circle passing through all three of its vertices) is simply half its hypotenuse, so