The Rational Zero Theorem- if f(x)=anxn+… a1x1+ a0 has integer coefficients, then
every rational zero of f has the following form:
p
= factors of constant (a0)
q
factors of lead coefficient (an)
List the possible rational zeros of f using the rational zero theorem
a. f(x)= x3 + 2x2 – 11x + 12
b. fIx)= 4x4 -x3 -3x2 +9x -10
Find all real zeros of f(x)= x3 -8x2 + 11x + 20
Step 1: list all possible rational zeros.
Step 2: graph function on calculator. Look for a matching zero from step 1.
Step 3: start synthetic division to see if it is a zero.
Find all real zeros of f(x) = 10x4-11x3 -42x2 +7x + 12
Step 1: List all possible rational zeros
Step 2: graph function on calculator. Look for matching zero from step 1
Start 3: Synthetic division.- You will have to perform synthetic division twice to get
to a quadratic since it is degree 4.
Descartes Rule of Signs- another way to narrow down zero search
 The number of positive real zeros of f equal the number of sign changes in
the sign of the coefficient of f(x) or is less than this by an even number.
 The number of negative real zeros of f is equal to the number of sign changes
of the coefficients of f(-x) or is less than this by an even number.
F(x)= x6 – 2x5 + 3x4 -10x3 -6x2 -8x -8