I was tutoring a student, and this question came up on a mock standardized test. Assume the student has learned advanced precalculus at a high school level and has access to a graphing calculator. The question is meant to be solved in a few minutes, tops.
Which of the following equations has exactly two non-real solutions?
(A) $\quad x^3-x^2+2x-2 = 0$
(B) $\quad x^3 -2x^2 -5x +6 = 0$
(C) $\quad x^4-7x^2+12=0$
(D) $\quad x^3-8x^2+11x+20=0$
(E) $\quad x^4-5x^3+x^2 +25x-30=0$
The first strategy that came to my mind is using Descartes's Rule of Signs, but that test seems to be inconclusive. A graphical approach seems to be another way to approach it, but that seems inefficient to me, and not entirely reliable in a test taking scenario- where you may fumble with manipulating window sizes and whatnot.
The answer is cited as (A), and indeed, the first answer choice can be easily factored as $(x^2+2)(x-1)$, which verifies it as a solution (note: I haven't checked the roots of the other polynomials). But again, I do not expect that factoring should have been the intended approach here.
Does anyone see a quick method to solve this, or is this a poorly designed question?
Edit: To followup on this question. Indeed, Gregory was correct in the comments. This question was from a unofficial SAT II Math practice exam. While the SAT II Math test does not require a calculator for any of its questions, it doesn't prohibit the use of one. It doesn't even ban computer algebra systems, like those on, say, the TI-89. Now, this particular practice test was constructed with (some) questions designed to train students to use the algebra software.
Thanks for the suggested solutions, though. I like the calculus methods, and I think my student will appreciate them as well.