1. Descartes’ Rule of Signs.
I. Descartes’ Rule of Signs.
A) If you look from term to term in a polynomial looking for sign changes (when one term is positive and the next one is negative (or vice-versa) …
1) The number of positive real zeros = the number of
changes in the signs. Or that value decreased by an
even number (6, 4, 2, 0 for example).
2) If you change the sign of “x” to “– x” and then count the
sign changes (remember: even –’s = +, odd –’s = –),
then the number of negative real zeros = the number of
even number (7, 5, 3, 1 for example)

2. Descartes’ Rule of Signs.
II. PNI table (Positive/Negative/Imaginary). Descartes rule.
A) All the positive + negative + imaginary = total solutions.
1) So if you know you have a certain amount of total
solutions, but don’t have enough pos and neg to
make that total, then the rest must be imaginary.
III. Extrema: Min/max values of the functions.
A) If you are checking all the possible solutions “k” using
synthetic substitution and …
1) k is a positive number AND the terms below the bar are all positive or
all negative #s, then k is an extrema and the graph will not curve again.
a) This means there are no more factors > than k.
2) k is a neg number AND all the terms below the bar alternate between
positive and negative ( etc.), then k is a lower extrema and the
graph will not curve again. (no more factors < than k).

3. Descartes’ Rule of Signs.
IV. Number of Solutions & Max number of curves.
A) Descartes’ rule of signs says that …
All the positive + negative + imaginary = total solutions. 1) The total # of solutions is equal to the degree of f(x).
2) The [degree – 1] is the maximum # of curves.
V. Approximating solutions using synthetic substitution.
A) If you are using synthetic substitution on two numbers, and the remainder of f(m) is a + # and the remainder of f(n) is a – #, then there is a solution between the two x-values.
1) Remember, the remainder is the y-value of f(#).
For example: If f(3) = -5 and f(4) = 2, then there is a solution between x = 3 and x = 4 on the graph, because for the y-value of the graph to go from negative (–5) to positive (2), it had to go through zero (a solution/root).

4. Example: PNI Chart: (Descartes Rule of Signs)
x6 – 2x5 + 3x4 – 10x3 – 6x2 – 8x – 8
– – – – – has 3 sign changes.
If you change it to (– x)n, it becomes.
(– x)6 – 2(– x)5 + 3(– x)4 – 10(– x)3 – 6(– x)2 – 8(– x) – 8
– –
which also has 3 sign changes
positive negative imaginary degree of f(x)
*Note: Only one of the above rows of PNI is the correct one. P N I
TOTAL Zeros 3 6 1 2 4