1. 6.5 Theorems About Roots of Polynomial Equations
6.5.1 Rational Root Theorem

2. 6.5.1: Rational Root Theorem
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:

3. Example 1: Finding Rational Roots
Find the rational roots of 3x3 – x2 – 15x + 5 = 0
Step 1: List the possible rational roots.
So the possibilities are:

4. Example 1 Continued 3x3 – x2 – 15x + 5 = 0
Step 2: Test each possible rational root.
3( )3 – ( )2 – 15( ) + 5 = 0 1 1 1
____________ -8
3( )3 – ( )2 – 15( ) + 5 = 0 -1 16 -1 -1
3( )3 – ( )2 – 15( ) + 5 = 0
1/3
1/3
1/3
3( )3 – ( )2 – 15( ) + 5 = 0
88/9
-1/3
-1/3
-1/3
3( )3 – ( )2 – 15( ) + 5 = 0
280 5 5 5
3( )3 – ( )2 – 15( ) + 5 = 0 -5 -5
-320 -5
3( )3 – ( )2 – 15( ) + 5 = 0
-80/9
5/3
5/3
5/3
3( )3 – ( )2 – 15( ) + 5 = 0 30
-5/3
-5/3
-5/3

5. Example 2: Using the Rational Root Theorem
Find the roots of 2x3 – x2 + 2x – 1 = 0
Step 1: List the possible rational roots.
Step 2: Test each possible rational root until you find a root
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)
+1, +1/2
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=

6. 6.5 Theorems About Roots of Polynomial Equations
6.5.2 Irrational Root & Imaginary Root Theorem

7. Irrational Root TheoremIf
is a root, then it’s conjugate
is also a root

8. Example3: Finding Irrational Roots
A polynomial equation with rational coefficients has the roots ___________ and ___________. Find two additional roots.
The additional roots are: ____________ and _______________

9. Imaginary Root TheoremIf
is a root, then it’s conjugate
is also a root

10. Example 4: Finding Imaginary Roots
A polynomial equation with real coefficients has the roots 2 + 9i and 7i. Find two additional roots.
The additional roots are:______ and _____

11. 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