5-5 Theorems About Roots of Polynomial Equations
Quick Review
The Rational Root Theorem gives a way to determine the possible roots of a polynomial equation
P
(
x
)
=
0
.
p open x close equals 0 . If the coefficients of
P
(
x
)
p open x close are all integers, then every root of the equation can be written in the form
p
q
,
p over q , comma where p is a factor of the constant term and q is a factor of the leading coefficient.
The Conjugate Root Theorem states that if
P
(
x
)
p open x close is a polynomial with rational coefficients, then irrational roots that have the form
a
+
b
eh plus square root of b and imaginary roots of
P
(
x
)
=
0
p open x close equals 0 come in conjugate pairs. Therefore, if
a
+
b
eh plus square root of b is an irrational root, where a and b are rational, then
a
−
b
eh minus square root of b is also a root. Likewise, if
a
+
b
i
eh plus b i is a root, where a and b are real and i is the imaginary unit, then
a
−
b
i
eh minus b i is also a root.
Descartes’ Rule of Signs gives a way to determine the possible number of positive and negative real roots by analyzing the signs of the coefficients. The number of positive real roots is equal to the number of sign changes in consecutive coefficients of
P
(
x
)
,
p open x close comma or is less than that by an even number. The number of negative real roots is equal to the number of sign changes in consecutive coefficients of
P
(
−
x
)
,
p open negative x close comma or is less than that by an even number.
Example
Find the rational roots of
P
(
x
)
=
0
if
P
(
x
)
=
2
x
3
−
4
x
2
−
10
x
+
12
.
p open x close equals 0 , if p open x close equals , 2 x cubed , minus , 4 x squared , minus 10 x plus 12 .
List the possible roots:
±
1
2
,
±
1
,
±
3
2
,
±
2
,
±
3
,
±
4
,
±
6
,
±
12
.
plus minus , 1 half , comma plus minus 1 comma plus minus , 3 halves , comma plus minus 2 comma plus minus 3 comma plus minus 4 comma plus minus 6 comma plus minus 12 . . Use synthetic division to test roots.
So
x
−
3
x minus 3 and
(
2
x
2
+
2
x
−
4
)
open 2 , x squared , plus 2 x minus 4 close are factors of
P
(
x
)
.
p open x close .
P
(
x
)
=
(
x
−
3
)
(
2
x
2
+
2
x
−
4
)
p open x close equals open x minus 3 close open 2 , x squared , plus 2 x minus 4 close
Factor the quadratic.
P
(
x
)
=
2
(
x
−
3
)
(
x
+
2
)
(
x
−
1
)
p open x close equals 2 open x minus 3 close open x plus 2 close open x minus 1 close
Solve
2
(
x
−
3
)
(
x
+
2
)
(
x
−
1
)
=
0
.
2 open x minus 3 close open x plus 2 close open x minus 1 close equals 0 .
x
=
3
,
x
=
−
2
,
or
x
=
1
x equals 3 comma x equals negative 2 comma or x equals 1
The rational roots are
3
,
−
2
,
3 comma negative 2 comma and 1.
Exercises
List the possible rational roots of
P
(
x
)
p open x close given by the Rational Root Theorem.
-
P
(
x
)
=
x
3
+
4
x
2
−
10
x
+
6
p open x close equals , x cubed , plus 4 , x squared , minus 10 x plus 6
-
P
(
x
)
=
3
x
3
−
x
2
−
7
x
+
2
p open x close equals , 3 x cubed , minus , x squared , minus 7 x plus 2
-
P
(
x
)
=
4
x
4
−
2
x
3
+
x
2
−
12
p open x close equals , 4 x to the fourth , minus , 2 x cubed , plus , x squared , minus 12
-
P
(
x
)
=
3
x
4
−
4
x
3
−
x
2
−
7
p open x close equals , 3 x to the fourth , minus , 4 x cubed , minus , x squared , minus 7
Find any rational roots of
P
(
x
)
.
p open x close .
-
P
(
x
)
=
x
3
+
2
x
2
+
4
x
+
21
p open x close equals , x cubed , plus 2 , x squared , plus 4 x plus 21
-
P
(
x
)
=
x
3
+
5
x
2
+
x
+
5
p open x close equals , x cubed , plus 5 , x squared , plus x plus 5
-
P
(
x
)
=
2
x
3
+
7
x
2
−
5
x
−
4
p open x close equals , 2 x cubed , plus , 7 x squared , minus 5 x minus 4
-
P
(
x
)
=
3
x
4
+
2
x
3
−
9
x
2
+
4
p open x close equals , 3 x to the fourth , plus , 2 x cubed , minus , 9 x squared , plus 4
A polynomial
P
(
x
)
p open x close has rational coefficients. Name additional roots of
P
(
x
)
p open x close given the following roots.
-
1
−
i
and
5
1 minus i , and , 5
-
5
+
3
and
−
2
5 plus square root of 3 . and , minus square root of 2
-
−
3
i
and
7
i
negative 3 i , and , 7 i
-
−
2
+
11
and
−
4
−
6
i
negative 2 plus square root of 11 . and , minus 4 minus 6 i
Write a polynomial function with the given roots.
- 7 and 10
-
−
3
and
5
i
negative 3 , and , 5 i
-
6
−
i
6 minus i
-
3
+
i
,
2
,
3 plus i comma 2 comma and
−
4
negative 4
Determine the possible number of positive real zeros and negative real zeros for each polynomial function given by Descartes’ Rule of Signs.
-
P
(
x
)
=
5
x
3
+
7
x
2
−
2
x
−
1
p open x close equals , 5 x cubed , plus , 7 x squared , minus 2 x minus 1
-
P
(
x
)
=
−
3
x
3
+
11
x
2
+
12
x
−
8
p open x close equals negative 3 , x cubed , plus , 11 x squared , plus 12 x minus 8
-
P
(
x
)
=
6
x
4
−
x
3
+
5
x
2
−
x
+
9
p open x close equals , 6 x to the fourth , minus , x cubed , plus 5 , x squared , minus x plus 9
-
P
(
x
)
=
−
x
4
−
3
x
3
+
8
x
2
+
2
x
−
14
p open x close equals negative , x to the fourth , minus , 3 x cubed , plus , 8 x squared , plus 2 x minus 14
