6.5 Theorems About Roots of
Polynomial Equations
6.5.1 Rational Root
Theorem
6.5.1: Rational Root Theorem
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To find rational roots of an equation, you
must divide the factors of the constant, by the
factors of the leading coefficient
Factors of the constant (p)
Factors of the leading coefficient (q)
Possibilities are:
p
q
Example 1: Finding Rational Roots
Find the rational roots of 3x3 – x2 – 15x + 5 = 0
Step 1: List the possible rational roots.
p : 1,5
q : 1,3
So the possibilities are:
1
5
 1, ,5,
3
3
Example 1 Continued
Step 2: Test each possible rational root.
3( 1
1
5
 1, ,5,
3
3
3x3 – x2 – 15x + 5 = 0
)3 – ( 1 )2 – 15( 1 ) + 5 = 0 ____________
-8
16
3( -1 )3 – ( -1 )2 – 15( -1 ) + 5 = 0 ____________
0
3( 1/3 )3 – ( 1/3 )2 – 15( 1/3 ) + 5 = 0 ____________
88/9
3( -1/3 )3 – (-1/3 )2 – 15( -1/3 ) + 5 = 0 ____________
280
3( 5 )3 – ( 5 )2 – 15( 5 ) + 5 = 0 ____________
-320
3( -5 )3 – ( -5 )2 – 15( -5 ) + 5 = 0 ____________
-80/9
3( 5/3 )3 – (5/3 )2 – 15( 5/3 ) + 5 = 0 ____________
30
3( -5/3 )3 – (-5/3 )2 – 15( -5/3 ) + 5 = 0 ____________
Example 2: Using the Rational Root
Theorem
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Find the roots of 2x3 – x2 + 2x – 1 = 0
Step 1: List the possible rational roots.
+1, +1/2
Step 2: Test each possible rational root
until you find a root
2(1)3 –(1)2 + 2(1) – 1=
2(-1)3 –(-1)2 + 2(-1) – 1=
2(1/2)3 –(1/2)2 + 2(1/2) – 1=
Step 3: Use synthetic division with the
root you found in Step 2
Step 4: Find the rest of the roots by
solving (use quadratic formula if
necessary)
6.5 Theorems About Roots of
Polynomial Equations
6.5.2 Irrational Root &
Imaginary Root
Theorem
Irrational Root Theorem
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If
a b
is a root, then it’s conjugate
is also a root
a b
Example3: Finding Irrational Roots
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A polynomial equation with rational
coefficients has the roots ___________
2  5 and
___________.
Find two additional roots.
 7
The additional roots are: ____________ and
_______________
Imaginary Root Theorem
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If
a  bi
is a root, then it’s conjugate
is also a root
a  bi
Example 4: Finding Imaginary Roots
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A polynomial equation with real coefficients
has the roots 2 + 9i and 7i. Find two
additional roots.
The additional roots are:______ and _____
Example 5: Writing a Polynomial
Equation from Its Roots
Find a third-degree
polynomial equation with
rational coefficients that has
roots of 3 and 1 + i.
Step 1: Find the other
imaginary root
Step 2: Write the roots in
factored form
Step 3: Multiply the factors
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