Use synthetic division to identify integer bounds of the real zeros. Find the least upper bound and the greatest lower bound guaranteed by the Upper and Lower Bounds of Zeros theorem.

Question

s(x)=4x^4+20x^3+15x^2−5x−4

Upper Bound:?

Lower Bound:?

Answer 1

Or equivalently, we can use the rational roots theorem, whose proof is based on these theorems...

p = {+-1,+-2,+-4}

q = {+-1,+-2,+-4}

p/q = {+-1,+-1/2,+-1/4,+-2,+-4}

is the set of rational solutions

x=-1: 4-20+15+5-4 = -16+15+5-4 = -1+5-4 = 0

x=1/2 : 4/16 + 20/8 + 15/4 - 5/2 - 4

= 1/4 + 10/4 + 15/4 - 5/2 - 4

= 1/4 + 10/4 + 15/4 - 10/4 - 16/4

is also zero

x=-1/2 : 4/16 - 20/8 + 15/4 + 5/2-4

= 1/4 - 10/4 + 15/4 + 10/4 - 4

is also zero

x=-4: 1024 - 1280 + 240 + 20 - 4

is also zero

greatest lower bound is -4

and the least upper bound is 1/2