Ch 4-4 The Rational Root Theorem
Obj: To use the Rational Root Thm to identify the possible rational roots, and determine the number of positive and
negative roots a polynomial has
Consider the following polynomial 𝑃(𝑥) = 2𝑥 3 − 𝑥 2 − 25 ….. If 𝑃(𝑥) = 0:
- How many roots (complex roots)?
- Of the roots, how many are rational?
We could use synthetic division to test possible roots…that could take time…and where do we
start?
𝑝
Translation: If a Rational Root exists, it can be obtained by using , where “𝑝” is the factors of
𝑞
the constant term, and “𝑞” is the factors of the Leading Coefficient.
Example 1: Given 𝑃(𝑥) = 2𝑥 3 − 𝑥 2 − 25, find all POSSIBLE roots.
𝑝  25:
𝑞  2:
Now…IF any rational roots exist…it would be one (or more) of the above. (Hint: Try 5/2 )
Once the polynomial
is depressed to a
quadratic…we have
several ways to find
the rest of the roots.
Descartes Rule of Signs – used to determine the possible numbers and combinations of positive
and negative real zeros, by counting sign changes:
𝑓(𝑥)−→ # 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑧𝑒𝑟𝑜𝑠
Or less by an even #
𝑓(−𝑥)−→ # 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑧𝑒𝑟𝑜𝑠
Ex. 3
𝑓(𝑥) = 𝑥 4 − 2𝑥 3 + 7𝑥 2 + 4𝑥 − 15
P
N
I
𝑓(−𝑥) =
For the function below:
a) Determine the number of possible positive and negative real zeros (make a chart)
𝑝
b) List all possible rational zeros (use )
𝑞
c) Given one of the zeros/roots, find the remaining zeros/roots
𝑓(𝑥 ) = 𝑥 3 + 4𝑥 2 − 2𝑥 + 15 𝐺𝑖𝑣𝑒𝑛: −5 𝑖𝑠 𝑎 𝑧𝑒𝑟𝑜
For the function below:
a) Determine the number of possible positive and negative real zeros (make a chart)
𝑝
b) List all possible rational zeros (use )
𝑞
𝑥 3 + 8𝑥 2 + 16𝑥 + 5 = 0
4-4: The Rational Root Thms
p.205/ 5-21 Odds Use the Hints” Worksheet