Polynomial equation with complex answers

[axon23]

x⁴+4x³-7x²-12x+24=0


Ive been working on this with my brother for a long time and we don't need to find an answer. we just need to reach a point when we can say that the solution is complex. Any ideas? Any nudge in the right direction with be great.


We tried completing squares, bruteforcing the answer and many other failed attempts

 

[Defennder]

What do you mean by "the solution is complex"? There are 4 roots to this. If you find the roots using some online calculators (essentially cheating), you'll see that 2 of them are real and other 2 are complex.

 

[axon23]

yea but what if i wanted to figure out the answer without a calculator. how would the process be done?

 

[tiny-tim]

points of inflection

x⁴+4x³-7x²-12x+24=0

Hi axon23!

Differentiate twice to find the points of inflection …

then you should be able to sketch the graph.

 

[snipez90]

I'm not sure what you mean by complex solution but it can be shown that there are no rational solutions by the Rational Root Theorem.

 

[HallsofIvy]

Set the derivative equal to 0 to find max and min points. Since this even power, if all minimum values are positive, the graph does not cross the x-axis and there are no real roots.