Algebra Know-It-ALL

440 Polynomial Equations in Real Numbers

Finding an upper bound

How can we find an upper bound for the roots of the polynomial equation under investiga-

tion? Here’s the equation again:

x^4 − 2 x^3 − 13 x^2 + 14 x+ 24 = 0

The coefficients and constant are 1, −2,−13, 14, and 24 in order of descending powers of x.

Let’s set up a synthetic division array for this equation:

#1− 2 − 13 14 24

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If we plug in a positive real number as a “test root,” we can tell if it’s an upper bound for the

real solution set by looking at the values we get in the last row. If we get a nonzero remainder

and none of the numbers in the last row are negative, then our “test root” is an upper bound.

Let’s take absolute values of the coefficients and constant in the equation and pick out the

largest. In this case, that’s 24. It seems reasonable that this might be larger than or equal to all

the real roots. Let’s input 24 and see what happens:

(^241) − 2 − 13 14 24

(^241) − 2 − 13 14 24

1####
(^241) − 2 − 13 14 24
24###
1####
(^241) − 2 − 13 14 24
24###
122# # #
24 1 − 2 − 13 14 24
24 528 # #
122# # #