The numbers in the last row alternate in sign, telling us that −24 is a lower bound for
the real solution set. The fact that the absolute values diverge rapidly from 0 suggests that our
input number is far smaller than necessary. We can try something larger, but still negative, and
do the synthetic division again. As long as we plug in negative “test roots” and get numbers in
the last line that alternate in sign, we know that we’re inputting lower bounds.
Narrowing the interval
We can reduce the size of the interval containing the real solutions by repeatedly testing posi-
tive numbers as upper bounds, and by repeatedly testing negative numbers as lower bounds.
Once we get a positive “test root” that produces a negative number anywhere in the bottom
line, or a negative “test root” that fails to cause the numbers in the bottom line to alternate in
sign, we know that we are in the interval containing the real roots.
Suppose we gradually reduce the positive “test root” and gradually increase the negative
“test root” in the synthetic division array
#1− 2 − 13 14 24
####
#####If we use integers for simplicity, we’ll eventually get down to a smallest upper bound, and we’ll
also get up to a greatest lower bound. At that point, we can test numbers between those bounds
to look for real roots. In this particular example, we’ll get a smallest upper bound of 5 and a
greatest lower bound of −4.
Rational roots
There’s a lengthy but straightforward process you can use to find all the rational roots of a
polynomial equation in the standard form
anxn+an-1xn-1+an-2xn-2+ · · +a 1 x+b= 0where a 1 ,a 2 ,a 3 , ... an are nonzero rational-number coefficients of the variable x,b is a nonzero
rational constant, and n is a positive integer greater than 3. If b= 0, then you can’t use the
process, but you will at least know that 0 is a root. If b≠ 0, then you can go through the fol-
lowing steps.
- Make certain that all the numbers a 1 ,a 2 ,a 3 , ... an, and b are integers. If that is not
the case, multiply the equation through by the smallest constant that will turn all the
numbersa 1 ,a 2 ,a 3 , ... an, and b into integers. - Find all the positive and negative integer factors of b, the stand-alone constant. Call
these by the general name m. - Find all the positive and negative integer factors of an, the coefficient of xn (sometimes
called the leading coefficient). Call these by the general name n. - Write down all the possible ratios m/n. Call them by the general name r.
Digging for Real Roots 443