P.o.D. – Factor the polynomial
completely.
1.) ℎ(𝑎) = 4𝑎3 − 4𝑎
2.) 𝑑(𝑥 ) = 18𝑥 3 + 57𝑥 2 − 21𝑥
3.) 𝑓(𝑥 ) = 𝑥 3 + 5𝑥 2 − 2𝑥 − 24
4.) The only zeros of a
polynomial function f are 6, 1,
and -2. Write a possible third
degree equation for f.
5.) Find the exact zeros of the
polynomial function
𝑔(𝑥 ) = 3𝑥 (𝑥 − 1)2 (𝑥 + 𝑒)
1.) 4a(a+1)(a-1)
2.) 3x(3x-1)(2x+7)
3.) (x+3)(x-2)(x+4)
4.) 𝑓(𝑥 ) = (𝑥 − 6)(𝑥 − 1)(𝑥 + 2) =
𝑥 3 − 5𝑥 2 − 8𝑥 + 12
5.) 0, 1, (1), -e
11-5: The Rational-Root Theorem
Learning Target(s): I can apply
the rational root theorem; graph
polynomial functions; estimate
zeros of polynomial functions
using graphs.
Review: Rational = Fractional &
Real
The Rational Root Theorem
(RRT):
If a polynomial has a rational
𝑝
root, , then p is a factor of the
𝑞
constant term and q is a factor
of the leading coefficient.
EX: List the possible rational
roots of 3𝑥 3 − 13𝑥 2 + 2𝑥 + 8 = 0.
Then determine the rational
roots.
Factors of p: ±1, ±2, ±4, ±8
Factors of q: ±1, ±3
𝑝
Possible Rational Roots ( ):
𝑞
1
2
4
8
±1, ± , ±2, ± , ±4, ± , ±8, ±
3
3
3
3
*We can graph the polynomial
to find the actual rational roots.
2
x=− , 1, 4.
3
EX: Find the exact values for the
roots of 𝑥 3 + 6𝑥 2 − 13𝑥 − 6 = 0
*Find a root using any known
method.
X=2  (x-2) is a factor.
Now divide the polynomial by
this factor.
(Show synthetic or long division
on the board)
𝑥 2 + 8𝑥 + 3 = 0
This is a quadratic, so now we
can solve it for our remaining
two roots.
−8 ± √64 − 4(1)(3)
𝑥=
=
2
−8 ± √52
=
2
−8 ± 2√13
=
2
−4 ± √13
The three roots are
2, −4 + √13, −4 − √13
Descartes Rule of Signs:
- Used to determine the
possible number of zeros in a
polynomial in descending
order.
o 1. The number of positive
real zeros is the same as the
number of sign changes in
f(x) or an even number
less.
o 2. The number of negative
real zeros is the same as the
number of sign changes in
f(-x) or an even number
less.
EX: Find the number of possible
positive real zeros and the
number of possible negative real
zeros for
𝑝(𝑥 ) = 24𝑥 4 − 𝑥 3 − 2𝑥 2 + 5𝑥 + 1 .
Then determine the rational
zeros.
𝑝(𝑥 ) = 24𝑥 4 − 𝑥 3 − 2𝑥 2 + 5𝑥 + 1
+1
+0
+1
+0
2 sign changes
2 or 0 possible positive real zeros
𝑝(−𝑥 ) = 24(−𝑥)4 − (−𝑥 )3 − 2(−𝑥 )2 + 5(−𝑥 ) + 1
= 24𝑥 4 + 𝑥 3 − 2𝑥 2 − 5𝑥 + 1
+0
+1
+0
+1
2 sign changes
2 or 0 possible negative real
zeros
Positive
Real
Negative Complex
Real
(Imaginary)
Zeros
Zeros
2
2
2
0
0
2
0
0
*Solve by graphing
Solutions
0
2
2
4
There were two negative
solutions and 0 positive
solutions, so 2 solutions must be
complex (imaginary).
It should be noted that neither
of these two negative solutions
are rational (can be written as a
fraction).
EX: A manufacturer produces
boxes for a calculator company.
The boxes have a volume of 240
cubic cm. Their height is 6cm
less than their width, while
their length is 1cm less than
twice their width. Find the
dimensions of such as box.
𝑤 = 𝑤𝑖𝑑𝑡ℎ, ℎ = 𝑤 − 6, 𝑙 = 2𝑤 − 1
𝑉 = 𝑙𝑤ℎ
240 = (2𝑤 − 1)(𝑤)(𝑤 − 6)
240 = (2𝑤 2 − 𝑤)(𝑤 − 6)
240 = 2𝑤 3 − 12𝑤 2 − 𝑤 2 + 6𝑤
2𝑤 3 − 13𝑤 2 + 6𝑤 − 240 = 0
Solve the equation by graphing
w=8
h=w-6=8-2=2
L=2w-1=2(8)-1=16-1=15
8 x 2 x 15
Do the following on your own.
a.) List the possible rational
roots of 2𝑥 3 + 3𝑥 2 − 8𝑥 + 3 =
0. Then determine the
actual rational roots.
b.) Find the number of
possible positive and
negative real zeros for
𝑓(𝑥 ) = 𝑥 3 + 7𝑥 2 + 7𝑥 − 15.
Then determine the
rational roots.
a.) 𝑝 = ±1, ±3; 𝑞 = ±1, ±2
𝑝
1 3
= ±1, ± , ± , ±3
𝑞
2 2
1
𝑥 = −3, , 1
2
b.) 𝑓(𝑥 ) = 𝑥 3 + 7𝑥 2 + 7𝑥 − 15
1 positive
𝑓(−𝑥 ) = −𝑥 3 + 7𝑥 2 − 7𝑥 − 15
2 or 0 negative
𝑥 = −5, −3, 1
EX: Apply the rational root
theorem to identify possible
roots of
𝑓(𝑥 ) = 3𝑥 4 − 10𝑥 2 − 8𝑥 + 15. Then
find all rational roots.
𝑝: ± 1, 3, 5, 15
𝑞: ± 1, 3
𝑝
1
5
: ± 1, , 3, 5, , 15
𝑞
3
3
After graphing, we can
determine that 1 is the only
rational root of f(x).
Upon completion of this lesson,
you should be able to:
1. List possible rational roots
of a polynomial.
2. Apply Descartes Rule of
Signs.
For more information, visit
https://www.youtube.com/watch?v=8y7cliO
Wzxw
HW Pg.763 3-5, 7, 12