6-5 Theorems About Roots of Polynomial Equations
Algebra 2, Belllman, Bragg, Charles, Handlin, Kennedy, Prentice Hall
1.
THE RATIONAL ROOT THEOREM
You have learned several methods for finding the roots of a polynomial
equation (e.g. graphing, factoring). Another method involves analyzing
one or more integer coefficients of the polynomial in the equation.
(Note: A polynomial equation has roots. A polynomial function has zeros.)
Consider the equivalent equations: x 3  5 x 2  2 x  24  0 and ( x  2)( x  3)( x  4)
which have  2, 3 and 4 as roots. Note that all roots are factors of the
constant 24 . In general, if the coefficients, including the constant term,
in a polynomial equation are integers, then any integer root of the
equation is a factor of the constant term.
n0
A similar pattern applies to rational roots. Consider the equivalent
1
2
2
3
3
4
1 2
2 3
equations 24 x 3  22 x 2  5 x  6  0 and ( x  )( x  )( x  )  0 which have  , , and
3
4
as roots.
The numerators: 1, 2, and 3 are all factors of the constant term 6 .
The denominators: 2, 3, and 4 are all factors of the leading coefficient 24.
The role that the constant term and leading coefficient play in
identifying the rational roots of a polynomial equation is expressed in the
following Rational Root Theorem.
RATIONAL ROOT THEOREM:
p
is in simplest form and is a rational root of the polynomial equation
q
an x n  a x n1  ...  a x  a  0 with integer coefficients then p must be a
n1
1
0
factor of a0 and q must be a factor of a n .
If
You can use the Rational Root Theorem to find any rational roots of a
polynomial equation with integer coefficients.
Examples:
Find the rational roots of each of the following.
1.
x 3  x 2  3x  3  0
2.
x3  4x 2  2x  8  0
Find all roots of each of the following.
3.
2x3  x 2  2x  1  0
4. 3x 3  x 2  x  1  0
2. IRRATIONAL ROOT THEOREM AND IMAGINARY ROOT THEOREM
Number pairs of the form a  b and a  b are called conjugates.
You can often use conjugates to find the irrational roots of a polynomial
equation.
IRRATIONAL ROOT THEOREM:
Let a and b be rational numbers and let b be an irrational number.
If a  b is a root of a polynomial equation with rational coefficients,
then the conjugate a  b also is a root.
Examples:
5. A polynomial equation with integer coefficients has the roots:
1 3 and  11 . Find two additional roots.
6. A polynomial equation with rational coefficients has the roots:
2  7 and 5 . Find two additional roots.
Number pairs of the form a  bi and a  bi are complex conjugates.
You can use complex conjugates to find and equation’s imaginary roots.
IMAGINARY ROOT THEOREM:
If the imaginary number a  bi is a root of a polynomial equation with real
coefficients, then the conjugate a  bi also is a root.
Examples:
7. A polynomial equation with integer coefficients has the roots 3  i and
2i . Find two additional roots.
8. A polynomial equation with real coefficients has the roots 3i and
 2  i . Find two additional roots.
You can often use the Irrational Root Theorem and the Imaginary Root
Theorem to write a polynomial equation if you know some of its roots.
Examples:
9. Write a third-degree polynomial equation with rational coefficients
that has roots 3 and 1  i .
10.
Find a third-degree polynomial equation with rational coefficients that
has roots 1 and 2  i .
11. Find a fourth-degree polynomial equation with rational coefficients that
has roots i and 2i .