Review and Examples of Using the Rational Root Theorem
Example 1
List the possible rational roots of x3 - x2 - 10x - 8 = 0. Then determine the rational roots.
p
According to the Rational Root Theorem, if q is a root of the equation, then p is a factor of 8 and q is a
factor of 1.
possible values of p:
possible values of q:
1, 2, 4, 8
1
p
possible rational roots, : 1, 2, 4, 8
q
You can use a graphing utility to narrow down
the possibilities. You know that all possible
rational roots fall in the domain -8 x 8.
So, set your x-axis viewing window at [-9, 9].
Graph the related function f(x) = x3 - x2 - 10x - 8.
A zero appears to occur at -2. Use synthetic
division to check that -2 is a zero.
-2
1
1
-1
-2
-3
-10
6
-4
-8
8
0
Thus, x3 - x2 - 10x - 8 = (x + 2)(x2 - 3x - 4). Factoring x2 - 3x - 4 yields (x - 4)(x + 1). The roots of x3 - x2 10x - 8 = 0 are -2, -1, and 4.
Example 2
Find the roots of x3 +6x2 + 10x + 3 = 0.
There are three complex roots. According to the Integral Root Theorem, the possible rational roots of the
equation are factors of 3. The possibilities are 3 and 1.
r
3
-3
1
1
1
6
9
3
10
3
37 114
1
0
There is a root at x = -3.
There is one root at x = -3. The depressed polynomial is x2 + 3x + 1. Use the Quadratic Formula to find
the other two roots.
x=
x=
x=
b2 - 4ac
2a
32 - 4(1)(1)
2(1)
5
-b
-3
-3
2
The three roots are -3,
-3 + 5
-3 - 5
, and
.
2
2
Example 3
Find the number of possible positive real zeros and the number of possible negative real zeros for x4
- 5x2 +4. Then determine the rational zeros.
To determine the number of possible positive real zeros, count the sign changes for the coefficients.
x4
f(x) =
5x2
-5
-
1
yes
+
4
4
yes
There are two changes. So, there are two or zero positive real zeros.
To determine the number of possible negative real zeros, find f(-x) and count the number of sign changes.
f(-x) =
f(-x) =
(-x)4
x4
1
-
5(-x)2
5x2
-5
yes
+
+
4
4
4
yes
There are two changes. So, there are two or zero negative real zeros.
Determine the possible zeros.
possible values of p: 1, 2, 4
possible values of q: 1
p
possible rational zeros, q: 1, 2, 4
Test the possible zeros using the synthetic division and the Remainder Theorem.
r
1
-1
2
-2
4
-4
1
0
1
1
1
1
1
1
1
-1
2
-2
4
-4
-5
-4
-4
-1
-1
11
11
0
-4
4
-2
2
44
-44
4
0
0
0
0
180
180
1 is a zero.
-1 is a zero.
2 is a zero.
-2 is a zero.
So, 1 and 2 are positive zeros and -1 and -2 are
negative zeros. Use a graphing utility to check
these zeros. You can use the zero function on
the CALC menu to verify that the zeros you
found are correct.
Example 4
MANUFACTURING The volume of a box of pasta must be 120 cubic inches. The box is
7 inches longer than it is wide and six times longer than it is tall. Find the dimensions of the box.
The formula for the volume of a rectangular prism is V = wh, where  is the length, w is the width, and h
is the height. From the given information,  = 6h and w = 6h - 7.
V
120
120
0
0
= wh
= (6h)(6h - 7)(h)
= 36h3 - 42h2
= 36h3 - 42h2 - 120
= 6h3 - 7h2 - 20
Divide each side by 6.
According to Descartes’ Rule of Signs, there is
one positive real root and zero negative real
roots. Use the graphing utility to graph the
related function V(h) = 6h3 - 7h2 - 20.
Using the Rational Root Theorem, the possible rational roots are 1, 2, 4, 5, 10, 20,
4 5 5 5 10
20
, , , ,
, and
. Use the Factor Theorem until the one zero is found.
3 2 3 6 3
3
V(h)
V(2)
V(2)
V(2)
= 6h3 - 7h2 - 20
= 6(2)3 - 7(2)2 - 20
= 48 - 28 - 20
=0
The height of the box of pasta is 2 inches, the length is 12 inches, and the width is 5 inches.
1 1 1 2
, , , ,
2 3 6 3