Question 1134149: Indentify the number of positive, negative, and imaginary roots, all rational roots , and the total amount of roots.
4x^4-3x^3+2x^2-x+1=0
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! Original function has 4 sign changes so 4,2,0 positive roots
f(-x)=x^4-3x^3+2x^2-x+1 or no sign changes so 0 negative roots
rational roots can be factors of 1 divided by factors of 4
so they can be +/- 1, 1/2 and 1/4. But no negative roots, so they can be +1/4,1/2,1
look at 1 first since easiest to check
1
/4 -3 +2 -1 1
--4----1-----3----2----3 doesn't work (4-3+2-1+1 doesn't equal 0)
1/4
4 -2 3/2 -1/4 15/16
1/2
4 -1 -1/2 -5/4 doesn't work
There are 0 rational roots, 0 positive roots and the 4 roots are all complex (2 conjugate pairs.) The graph should not touch the x-axis.
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