Algebra II/Trig Honors
Unit 2 Day 6: Find Rational Zeros
Objective: Find all real zeros of polynomial functions
The polynomial function f x   64 x 3  152 x 2  62 x  105 has 
5
3
7
,  , and as its zeros.
8
2
4
Notice the numerators, ______________________, are factors of the constant term, -105.
Notice the denominators, _____________________, are factors of the leading coefficient, 64.
Rational Zero Theorem (also called the Rational Root Theorem)
If f x   a n x n      a1 x  a0 has integer coefficients, then every rational zero has the following form:
_____ = _____________________________________
Example 1: List possible rational zeros
List the possible rational zeros of f using the rational zero theorem.
a. f x   x 3  2 x 2  11x  12
b. f x   4 x 4  x 3  3x 2  9 x  10
Verifying Zeros – since we only listed possible zeros, we will need to find the actual zeros. We can do
this using ______________________________.
Example 2: Find zeros when the leading coefficient is 1
Find all zeros of f x   x 3  8x 2  11x  20
Step 1: List all possible rational zeros.
Step 2: Test these zeros using
synthetic division.
Step 3: Factor the trinomial in f(x)
and use the factor theorem.
If the leading coefficient is not 1, you can end up with a pretty long list of possible zeros. To eliminate
some of the possibilities, you can sketch a graph of the function.
Example 3: Find zeros when the leading coefficient is not 1
Find all real zeros of f x   10 x 4  11x 3  42 x 2  7 x  12
Step 1: List the possible rational zeros.
Step 2: Choose reasonable values from
the list above. Check using the graph of
the function.
Step 3: Check the values using synthetic
division until a zero is found.
Step 4: Factor out a binomial using the
result of the synthetic division.
Step 5: Repeat the steps above for the
remaining polynomial.
Step 6: Find the remaining zeros by
solving.
Example 4: Solve a multi-step problem
Some ice sculptures are made by filling a mold with water and then freezing it. You are making such an
ice sculpture for a school dance. It is to be shaped like a pyramid with a height that is 1 foot greater than
the length of each side of its square base. The volume of the ice sculpture is 4 cubic feet. What are the
dimensions of the mold?
HW: Page 132 #4-36 (M4), 38, 41-44, 45, 48