Question 1192454
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Plug in x = 0.5
P(x)=6x^4-6x^2+4x-2
P(0.5)=6(0.5)^4-6(0.5)^2+4(0.5)-2
P(0.5)=-1.125
The result is negative.


Repeat for x = 1
P(x)=6x^4-6x^2+4x-2
P(1)=6(1)^4-6(1)^2+4(1)-2
P(1)=2
The result is positive.


The actual numeric values don't matter. 
All we care about are the signs of the results.


The value of P(0.5) is negative and P(1) is positive.


The transition from negative to positive means that somewhere along the line is at least one x intercept or root. 
This is because the curve is continuous meaning there aren't any gaps or jumps.
Any polynomial is continuous.


Using a graphing calculator like desmos or geogebra would show that the root on this interval is roughly x = 0.83
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