Find the smallest positive integer and the largest
negative integer that, by the Upper- and Lower-Bound
Theorem, are upper and lower bounds for the real zeros
of the polynomial function.
P(x) = 2x^3 + x^2 - 25x + 9
If we do synthetic division with a
positive number and find that all
the signs on the bottom row of the
synthetic division are non-negative,
then the positive number that we are
doing the synthetic division with is
an upper bound for all the real zeros.
We try 1 for an upper bound:
1 | 2 1 -25 9
| 2 3 -22
2 3 -22 -13
No the bottom row of numbers are not
all non-negative.
We try 2 for an upper bound:
2 | 2 1 -25 9
| 4 10 -30
2 5 -15 -21
No the bottom row of numbers are not
all non-negative.
We try 3 for an upper bound:
3 | 2 1 -25 9
| 6 21 -12
2 7 -4 -3
No the bottom row of numbers are not
all non-negative.
We try 4 for an upper bound:
4 | 2 1 -25 9
| 8 36 44
2 9 11 53
Yes the bottom row of numbers are all
non-negative.
Therefore 4 is the smallest positive integer that,
by the Upper- and Lower-Bound Theorem, is an upper
bound for the real zeros of the polynomial function.
--------------------------------------
If we do synthetic division with a
negative number and find that the
signs on the bottom row of the
synthetic division alternate in sign,
then the negative number that we are
doing the synthetic division with is
a lower bound for all the real zeros.
[In this process we can consider 0
on the bottom row to have either sign].
We try -1 for a lower bound:
-1 | 2 1 -25 9
| -1 0 25
2 0 -25 33
No the bottom row of numbers do not
alternate in sign.
We try -2 for a lower bound:
-2 | 2 1 -25 9
| -4 6 38
2 -3 -19 47
No the bottom row of numbers do not
alternate in sign.
We try -3 for a lower bound:
-3 | 2 1 -25 9
| -6 15 30
2 -5 -10 39
No the bottom row of numbers do not
alternate in sign.
We try -4 for a lower bound:
-4 | 2 1 -25 9
| -8 28 -12
2 -7 3 -3
Yes the bottom row of numbers alternate
in sign, so
Therefore -4 is the largest negative integer that, by
the Upper- and Lower-Bound Theorem, is a lower bound
for the real zeros of the polynomial function.
Edwin