6.5 Theorems About Roots of Polynomial Equations 6.5.1 Rational Root Theorem

6.5 Theorems About Roots of Polynomial Equations

6.5.1 Rational Root Theorem

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:

q

p

Example 1: Finding Rational Roots

Find the rational roots of 3x3 – x2 – 15x + 5 = 0

5,1: p

Step 1: List the possible rational roots.

3,1: qSo the possibilities are:

3

5,5,

3

1,1

Example 1 Continued

Step 2: Test each possible rational root. 3x3 – x2 – 15x + 5 = 0

3( )3 – ( )2 – 15( ) + 5 = 0

3( )3 – ( )2 – 15( ) + 5 = 0

3

5,5,

3

1,1

3( )3 – ( )2 – 15( ) + 5 = 0

3( )3 – ( )2 – 15( ) + 5 = 0

3( )3 – ( )2 – 15( ) + 5 = 0

3( )3 – ( )2 – 15( ) + 5 = 0

3( )3 – ( )2 – 15( ) + 5 = 0

3( )3 – ( )2 – 15( ) + 5 = 0

1 1 1

-1 -1 -1

1/3 1/3 1/3

-1/3 -1/3 -1/3

5 5 5

-5 -5 -5

5/3 5/3 5/3

-5/3 -5/3 -5/3

____________

____________

____________

____________

____________

____________

____________

____________

-8

16

0

88/9

280

-320

-80/9

30

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=

6.5 Theorems About Roots of Polynomial Equations

6.5.2 Irrational Root & Imaginary Root Theorem

Irrational Root Theorem

If

is a root, then it’s conjugate

is also a root

ba ba

Example3: Finding Irrational Roots

A polynomial equation with rational coefficients has the roots ___________ and ___________. Find two additional roots.

The additional roots are: ____________ and _______________

52 7

Imaginary Root Theorem

If

is a root, then it’s conjugate

is also a root

bia bia

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 _____

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