Can some please explain to me upper and lower bound theorems?

Question

You can even state sites that can really explain well

Answer 1

are you talking about the one involving synthetic division?

in that form

if the 3rd row numbers are all positive , and the test root is positive, then it is already an upper bound. since choosing a bigger number will make the resulting values on the 3rd row even bigger. thus you wont expect it to go back to 0.

if the 3rd row numbers alternate in signs, and the test root is negative,

then it is another bound. this would be the lower bound. you need to choose a new test root from higher numbers.