Righto, i need help with this question my friend who's sitting for STPM gave me. OK..i am egoistical, and i refuse to admit i can't solve it.....but after an hour of trying to figure it out..have given up.
How do you prove a given quartic equation has 4 imaginary roots?
x^4 + 3x^2 - 2x + 2=0
What i did was this:
I first calculated the squares of the roots,and it was negative as expected. But thats not conclusive enough. One possiblility is that there could be two complementary imaginary roots that just happen to be larger than the other two real roots. I dunno if its possible to use the Newton-Rhapson method......its not exactly proving....and is kinda messy i think. So, tell me how to prove this without actually calculating the roots.
I got another method from Tim Weng. Find the turning points, and using them, try t draw and determine wether the entire graph lies above or below the x-axis. I tried it, but got stuck on the graph drawing as there are only 2 turning points, instead of the 3 that quartic graphs should have. Fluctuation point?
In the end, it seems like this all boils down to graph sketching.....which i wanted to avoid doing. Any other way to solve this question?
Cheers Mates!
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