Algebra 2: Section 6

5 Theorems about Roots of Polynomial Equations
Standard: Students demonstrate knowledge of how real and complex numbers are
related arithmetically.
Conjugates are:
number pairs of the form
an
d
.
Complex conjugates are:
number pairs of the form
and
.
Quick Check
Ex. 1)
a)
A polynomial equation with rational coefficients has the roots
and
. Find two additional roots.
b) One of the roots of a polynomial equation is
. Can you be certain
that
is a root of the equation? Explain.
Rational Root Theorem
If
is in simplest form and is a rational root of the polynomial
equation
with integer coefficients,
then
p
must be a
factor
of
and
q
must be a
factor
of
.
Irrational Root Theorem
Let
a
and
b
be rational numbers and let
be an irrational
number. If
is a root of a pol
ynomial equ
ation with rational
coefficients, then the conjugate
also is a root.
Imaginary Root Theorem
If the imaginary number
is a root of a polynomial equation
with real coefficients, then the conjugate
also is a root.
Ex. 2)
Find the roots of each equation.
a)
b)
Ex. 3)
a)
If a polynomial equation with real coefficients has
and
among its roots, then what two other roots must it have?
b)
Critical T
hinking
Describe the degree of the equation.
Assignment page 345 #1

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