1. Remainder Theorem
If P(x) is divided by x – r, then the remainder is equal to P(r).
We can find P(r) using Synthetic Division.

2. Factor Theorem If P(r)= 0, then x – r is a factor of P(x).
If x – r is a factor of P(x), then P(r) = 0
Reminder: P(r) is the remainder of P(x) divided by x – r.

3. Properties of Polynomials
An nth degree polynomial has n zeros. The zeros could be complex.
Ex) P(x) = 2x³ - 4x² + 2x zeros
Ex) P(x) = 3x x zeros
(Polynomial Factorization Theorem)
An nth degree polynomial has n linear factors.
Ex) P(x)= x4 – 8x³+ 14x²+ 8x -15 =
( x -1)(x+1)(x -3)(x-5)

4. Conjugate Pairs Theorem
Let P(x) be an nth degree polynomial.
If a+bi is a zero of P(x), then the conjugate a – bi must also be a zero of P(x).
Ex) Let P(x) = x² - 4x +5
If P(2 + i) = 0, then P(2 – i) = 0
Ex) Let P(x) = x³ + 2x² +x +2
P(i) = 0, P(-i) = 0, P( -2) = 0

5. Descartes Rule of Signs
Let P(x) be a polynomial of the form
P(x) = anxn+an-1xn-1+…..a1x+a0
The number of positive real zeros of P(x) is equal to the number of sign changes of P(x) or is less than that number by an even integer.
The number of negative real zeros of P(x) is equal to the number of sign changes in P(-x) or is less than that number by an even integer.

6. Example Find all possible positive, negative real and nonreal zeros of
P(x) = 4x4- 3x³ +5x² + x – 5

7. Rational Zero Theorem Let P(x) = anxn+an-1xn-1+…..a1x+a0
If P(x) has rational zeros, they will be of the form p/q, where
p is a factor of a0 , and
q is a factor of an

8. Example
Find the list of all possible rational zeros for each function below.
A) P(x) = x³ + 3x² - 8x + 16
B) P(x) = 3x4 + 14x³ - 6x² +x -12
C) P(x) = 2x³ - 3x² + x – 6

9. Factoring for the finding Zeros of Polynomials
For 2nd degree, we factored or used the quadratic formula.
x² - 3x – 15 = 0 , ( x – 5)(x + 2) = 0 so
x = 5 or x = -2.
For 3rd degree, we factored.
x³ - x² - 4x + 4 = 0 , x²(x -1) -4(x – 1) =0
( x – 1)(x² - 4) = 0 , (x – 1)(x - 2)(x + 2) =0
x = 1, x = 2, x = -2
But, Factoring by traditional means doesn’t always work for all polynomials.

10. Strategy for Finding all the zeros of a Polynomial
Step 1: Use Descartes Rule of Signs
Step 2: Use Rational Zeros Theorem to get list of possible rational zeros.
Step 3: From the list above, test which ones make P(x) = 0.
Do this using SYNTHETIC DIVISION!!!!
Do not plug in the values into P(x)!!!
We want to factor P(x) until we get a quadratic function.