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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)

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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:


Example 1 finding rational roots
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:


Example 1 continued
Example 1 Continued

3x3 – x2 – 15x + 5 = 0

Step 2: Test each possible rational root.

3( )3 – ( )2 – 15( ) + 5 = 0

1

-8

1

1

____________

____________

____________

____________

____________

____________

____________

____________

3( )3 – ( )2 – 15( ) + 5 = 0

-1

16

-1

-1

0

3( )3 – ( )2 – 15( ) + 5 = 0

1/3

1/3

1/3

88/9

3( )3 – ( )2 – 15( ) + 5 = 0

-1/3

-1/3

-1/3

280

3( )3 – ( )2 – 15( ) + 5 = 0

5

5

5

3( )3 – ( )2 – 15( ) + 5 = 0

-320

-5

-5

-5

-80/9

3( )3 – ( )2 – 15( ) + 5 = 0

5/3

5/3

5/3

3( )3 – ( )2 – 15( ) + 5 = 0

30

-5/3

-5/3

-5/3


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=

Example 2: Using the Rational Root Theorem


6 5 theorems about roots of polynomial equations1

6.5 Theorems About Roots of Polynomial Equations

6.5.2 Irrational Root & Imaginary Root Theorem


Irrational root theorem
Irrational Root Theorem

  • If

    is a root, then it’s conjugate

    is also a root


Example3 finding irrational roots
Example3: Finding Irrational Roots

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

    The additional roots are: ____________ and _______________


Imaginary root theorem
Imaginary Root Theorem

  • If

    is a root, then it’s conjugate

    is also a root


Example 4 finding imaginary roots
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

Example 5: Writing a Polynomial Equation from Its Roots


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