1 / 11

6.5 Theorems About Roots of Polynomial Equations

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)

sohalia-bijoy
Download Presentation

6.5 Theorems About Roots of Polynomial Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 -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

  5. 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. 6.5 Theorems About Roots of Polynomial Equations 6.5.2 Irrational Root & Imaginary Root Theorem

  7. Irrational Root Theorem • If 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 Theorem • If 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. 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

More Related