Solve:
by the remainder theorem, we see that is a root. thus by the factor theorem, is a factor.
dividing our cubic by , we see that:
thus, we have or
i leave this part to you. we found our roots to be and , so we in fact can split up the real line into the intervals , , and
on which of these intervals is the cubic positive?
By inspection is a factor because is a zero of the expression
then by use synthetic division
Compare coefficients of the highest order term to get a = 1 and lowest order to get c = 4 first because it is easy
by comparing the coefficient of the term with x^2 or otherwise you should easily get b = -4
for the inequality you must sketch a graph.
You know be able to tell that for large positive values of x the function will be positive, and for large negative values of x the function will be negative.
You should also know the the sign of the function can only change at a root.
you should get a graph like this. the image below.
Remember the inequality is greater than zero not greater than or equal to, don't be lazy and just write x >1