While the following process is ostensibly to find the least upper and greatest lower integral bounds for the real roots of polynomial equations, it has a nice side benefit  pairs of consecutive integers between which a real root is located are also uncovered while locating these bounds. For this reason, I will point out both bounds and pairs of integers that 'trap' real roots in the following examples.
Background Information: As you might have already surmised, finding the roots (or solutions) of an equation is usually not a trivial process. At the moment we are concerned with finding the roots (or solutions) of a polynomial equation. Now, polynomials are one of the most basic algebraic expressions since there are no exponentials, logarithms, fractions with variables in the denominator, or variables under radicals. (a polynomial is an algebraic expression in which the exponents of the variables are whole numbers and no variables are in the denominator) However, finding the solutions of polynomial equations is not easy. Solutions to polynomial equations can be either real numbers or imaginary numbers. Right now, we are going to concern ourselves with just the real solutions. Recall that if the real number r is a solution to the polynomial equation P(x) = 0 then P(r) = 0. Since r is a real number and P(r) = 0, then y = P(x) must touch the xaxis where x = r
Here is an example: let P(x) = 5x^{4}  3x^{3}  x^{2} + x  2 P(1) = 0 and 1 is a real number so the graph of y = 5x^4  3x^3  x^2 + x  2 will touch the xaxis where x = 1.
Notice that we say the graph touches the xaxis at x = r. That means it may either pass through the axis or not, but the graph will come into contact with the xaxis.
Any old bounds won't do: At times it is useful to find bounds for the real roots of the polynomial. That means we are looking for a number that is greater than all roots (an upper bound) and a number that is less than all roots (a lower bound). That is actually pretty easy for the polynomials which we will encounter in this class because I could just pick a really big number, say 1000000, and figure that no root will be bigger than 1 million. Likewise, I can guess than no root will be less than 1000000 and will probably be right. But knowing that the graph will touch the xaxis somewhere in the interval ( 1000000, 1000000) doesn't really give much useful information about the location of the roots. We would like the location of the roots to be made with a bit more specificity.
LEAST UPPER BOUND AND GREATEST LOWER BOUNDS What would really help is if we could determine the SMALLEST upper bound and the GREATEST lower bound so that we have the roots confined to the smallest interval possible. If the roots happen to be irrational then there is no smallest upper bound because the root itself is a nonterminating, nonrepeating decimal so we can't find a number that will lie right next to it on the number line. So, we compromise and look for the LEAST INTEGER upper bound (LUB) and the GREATEST INTEGER lower bound (GLB) that we can find. (Recall that integers are {.....3, 2, 1, 0, 1, 2, 3,....})
To help us in the finding the bounds on the real roots we will make use of the theorem in your text:
We use this theorem in a very systematic way to find the first positive integer that satifies the condition for an upper bound. That is, we try 1, then 2, then 3 (and so on) until we find the first positive integer that works. To find the lower bound we do the same thing. Try 1, then 2, then 3 (and so on) until we find the first negative integer that satisfies the condition for a lower bound.
Here is an example: Determine the least integral upper bound and greatest integral lower bound for the real roots of the polynomial x^{4}  3 x^{2} + x  12= 0 As was previously stated, we will use synthetic division to first try 1, then 2, and so on until we find the FIRST POSITIVE INTEGER that passes the upper bound test. To consolidate the synthetic division, only the third lines of each synthetic division step will be shown Since the last line contains no negative numbers we can declare that 3 is the least integral upper bound that we can find using this theorem. Now we will look for the greatest lower bound. In this case, we are looking for the FIRST NEGATIVE INTEGER that passes the upper bound test. We will try,  1, 2, and so on until the last line of the synthetic division is alternating in sign. Again, to consolidate the synthetic division, only the third lines of each synthetic division step will be shown Since the last line is alternating positive and negative we can declare that  3 is the greatest integral lower bound that we can find using this theorem. NOTE: It is not always the case that the upper and lower bounds will be additive inverses of one another.
LOCATING REAL ROOTS BETWEEN CONSECUTIVE INTEGERS We can glean another piece of useful information from these two tables. For that piece, we will look to the Location Theorem.
From the tables we can see the following: P(0) =  12 P(2) and P(3) have opposite signs therefore according to the Location Theorem, the equation P(x) = 0 has at least one REAL root between 2 and 3. There is also a real root between 3 and  2. Note: We are generally interested in locating real roots between consecutive integers for two reasons.
One last point: Why don't we find bounds to imaginary roots as well as the real roots? That is because imaginary numbers do not have the order property that applies to the set of real numbers That means that the concept of < and > has no meaning for imaginary numbers. It might be nice if we could say that all roots are < the LEAST UPPER BOUND (LUB) but that statement has no meaning for imaginary roots. So, the best we can say is that all REAL roots are < the LUB and all REAL roots are > the GLB. Now you have to admit, this IS interesting stuff, isn't it? © 1999 Jo Steig
