# Algebra 2: Section 6-5 Theorems about Roots of Polynomial Equations

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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

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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