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6-5 Part II Notes: Theorems about Roots of Polynomial Equations
Conjugates: Number pairs in the form a  b , a  b
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 is also a root.
Example:
A polynomial equation with rational coefficients has the roots 2  5 and
Find two additional roots.
7.
By the Irrational Root Theorem:
If 2  5 is a root, then __________ is a root.
If
7 is a root, then ________ is a root.
Complex Conjugates:
Number pairs in the form a  bi , a  bi
Imaginary Root Theorem:
If the imaginary number a + bi is a root of a polynomial equation with real coefficients,
then the conjugate a – bi is also a root.
Example:
A polynomial equation with real coefficients has the roots 2 + 9i and 7i. Find 2
additional roots.
By the Imaginary Root Theorem:
If 2 + 9i is a root, then __________ is also a root.
If 7i is a root, then ________ is also a root.
Example:
Find a third degree polynomial equation with rational coefficients that has roots
-2, and 2-i.
Step 1:
_________ is also a root.
Step 2:
Write the equation in factored form.
Step 3:
Distribute.