1. Zeroes of Polynomial Functions
I. The Fundamental Theorem of Algebra.
A) If a poly has degree n, then it has n solutions.
1) Any solution repeated twice counts as two solutions.
Any repeated 3 times counts as 3 solutions. Etc.
Example: 7x3 + … has 3 solutions. –9x17 + … has 17 sol.
II. Conjugate pairs: two answers that are opposites of each
other. (one + , one –)
1) Complex conjugates theorem: If you have one
imaginary solution, you will also have its conjugate.
( a + bi and a – bi )
2) Irrational conjugates theorem: If you have one
irrational solution, you will also have its conjugate.
( a + √ b and a – √ b )

2. Zeroes of Polynomial Functions
III. The Rational Zero Theorem.
A) If a poly has integer coefficients, then all the rational zeros must be…
1) ± factors of the last term (the constant)
factors of the first term (coefficient)
B) To list all those factors…
1) Write each top # over the 1st bottom #.
2) Then write each top # over the 2nd bottom #.
3) Repeat for each bottom #.
4) Reduce fractions.
5) Do not write the same number twice.

3. Zeroes of Polynomial Functions
IV. Writing factors from zeros (and conjugate pairs).
A) Zeros are written in the form (x – h).
1) Where “h” is the zero. (Change the sign)
B) Remember: conjugates come in pairs. So if you have one imaginary zero, you have another of the opposite sign. Same with radicals.
Example: zeros are 6, – 8, 2i write as a factored polynomial.
y = (x – 6) (x + 8) (x – 2i) (x + 2i) (the i’s are conjugate pairs)
C) If you have the factors and want the zeroes, set each factor equal to zero and solve for x.
Example: find the zeros of f(x) = (x + 4) (x – 3) (2x – 7)
x + 4 = x – 3 = x – 7 = 0
x = x = x = 7/2

4. Zeroes of Polynomial Functions
V. 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)

5. Zeroes of Polynomial Functions
VI. PNI table (Positive/Negative/Imaginary). Descartes rule.
A) All the positive + negative + imaginary = total.
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.
VII. 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.

6. Zeroes of Polynomial Functions
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) P N I
TOTAL Zeros 3 6 1 2 4