Question 1088659
if p(x) = x^4+3x^2-7x+1, we need to find p(a) = 0.

according to the Fundamental algebra theorem, there must be 4 roots on the complex number.

+-1 could be a rational root.

p(1) = -2 and p(-1) = 12, so the polynomial does not have any rational root.

but using the intermediate value theorem p(1)*p(-1) = -24<0 so there's at least a root between (-1,1), beacuse the images has different sign, so must be a root between the two pre-images. That solve the question.

****bonus: p(2) = 15, and as before p(1)*(p2) = -30<0 so there's at least a root between (1,2).

Using derivative and a software, we can see that there's 2 real roots:

x1= 0,15296291586997 and x2= 1,31770465653627 which are irrational numbers and are consistent with intermediate value theorem intervals, the other two root are complex and cannot be calculated with my software.*****

@natolino_