1. The Rational Zero Theorem
Let f(x) = anxn + an-1xn-1 + … + a1x + a0 represent a polynomial function with integer coefficients.
If p/q in lowest terms is a rational zero of f, then p must be a factor of a0 (constant or last #) and q is a factor of an (leading coefficient).

2. a) State all possible rational zeros for g(x) = x3 + 2x2 - 3x + 5
Example 4
Factors of the constant
Factors of the leading coefficient
a) State all possible rational zeros for g(x) = x3 + 2x2 - 3x + 5
+5,+1 +1
Possible rational zeros: +5, +1
b) State all possible rational zeros for g(x) = 6x3 + 6x2 - 15x - 2
+2,+1
+6,+3,+2,+1
+2,+1
Possible rational zeros:
+6,+3,+2,+1

3. Corollary
If the coefficients of a polynomial function are integers such that an = 1 and a0 = 0, any rational zeros of the function must be factors of a0.

4. Try Check Your Progress #1A
Answer

5. Example Find all the zeros of the function.
From the corollary to the Fundamental Theorem of Algebra – we know that there are exactly 4 complex roots.
Using Descartes Rule of signs, there are 4, 2, or 0 positive real roots and 0 negative real roots.
Use the Rational Zeros Theorem:
Test some of these and find that f(5) = 0
Depressed polynomial
Factor
Zeros are 5, 3/2, 2i, and –2i

6. Homework Assignment #39
p odd, 32-34, 44-45