Find the equation which has 4 distinct roots 
For which value of $k$ does the equation $x^4 - 4x^2 + x + k = 0 $ have four distinct real roots?

I found this question on a standardized test, and the answer presumably  relies on a graphing calculator. Is there some method to solve this problem without the aid of a calculator?

Heck, I may as well just put this down as an "answer" (for which your question does not have just one)...
Notice that there is no cubic term:  this tells us that the sum of the four roots is zero.  The value of $ \ k \ $ is the product of these four roots.  With the four roots being distinct, we must also have the sum of all six distinct products of pairs of the roots equal -4 (the quadratic coefficient) , and the sum of all four distinct products of three roots equal -1 (the negative of the linear coefficient). [BTW, was this a multiple-choice question?  I ask because there is not a unique value for $ \ k \ $... ]
To clarify the above block of text a bit, consider your four roots to be $ \ p \ , \ q \ , \ r \ , \ \text{and} \ \ s \ $ .  The factored form of the polynomial is then $ \ (x - p) \cdot (x - q) \cdot (x - r) \cdot (x - s) \ \ $ ; consider what happens once you multiply this out into a quartic polynomial.  Among the choices you show, only $ \ k = 1 \ $ is correct.
