Question 1088659: Please help to solve that there is a root of the equation , using intermediate value theorem.
Answer by natolino_2017(77)   (Show Source):
You can put this solution on YOUR website! 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_
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