# Finding the bounds on the real zerosFind the bounds on the real zeros of the following function. Please show all of your work.f(x)=x^4+2x^3-x^2-1

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embizze | Certified Educator

We are asked to find the bounds on the real zeros of `f(x)=x^4+2x^3-x^2-1`

The following rule helps to find the bounds on the real zeros: When using synthetic division (see reference) if the result is all positive, then the divisor is an upper bound on the real roots. Also, if the result alternates in sign (nonnegative-nonpositive-nonnegative etc...) then the divisor is a lower bound on the real zeros.

The results using synthetic division with a given divisor `d` :

d=-3: 1 -1 2 -6 17  Thus -3 is a lower bound for the real roots.

d=-2: 1 0 -1 2 -5

d=-1: 1 1 -2 2 -3

d=0: 1 2 -1 0 -1

d=1: 1 3 2 2 1 Thus 1 is an upper bound on the real roots.

Thus the real roots lie in (-3,1). ( -3<r<1 )

** The two real roots are approximately .84449855 and -2.470979

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