24 1 − 2 − 13 14 24
24 528 # #
1 22 515 # #24 1 − 2 − 13 14 24
24 528 12,360 #
1 22 515 # #24 1 − 2 − 13 14 24
24 528 12,360 #
1 22 515 12,374 #(^241) − 2 − 13 14 24
24 528 12,360 296,976
1 22 515 12,374 #
(^241) − 2 − 13 14 24
24 528 12,360 296,976
1 22 515 12,374 297,000
None of the numbers in the bottom row are negative. Therefore, 24 is an upper bound for
the real solution set. The fact that the numbers increase so fast (we might even say that they
“blow up”) suggests that 24 is a much larger than the largest root of the equation. We can try
something smaller, but still positive, and do the synthetic division again. As long as we input
positive “test roots” and never see a negative number in the last row, we know that we’re input-
ting upper bounds.
Finding a lower bound
If we plug in a negative number as a “test root,” grind out the synthetic division process, get
a nonzero remainder, and discover that the numbers in the last row alternate between positive
and negative, it tells us that we’ve found a lower bound for the real solution set. Let’s take the
absolute values of the coefficients and constant, and pick out the negative of the largest result.
That’s −24. It’s a good bet that this is smaller than all the real roots. Let’s input it to the array
and find out:
− 24 1 − 2 − 13 14 24
Digging for Real Roots 441