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Mastering Polynomials: Factoring Tools, Theorems & Techniques

Learn to solve and graph polynomials efficiently using factoring tools like Long Division and Synthetic Division. Discover the Rational Root Theorem and methods to find possible rational roots. Dive deep into the Irrational and Imaginary Root Theorems, and explore the Fundamental Theorem of Algebra. Brush up your skills with examples and exercises to ace your upcoming assessments.

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Mastering Polynomials: Factoring Tools, Theorems & Techniques

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  1. Chapter 6Final Touches

  2. Pg 325 #42-47 Determine whether each binomial is a factor of Reminds me of a “bank” Limits the ones I have to check! Of course a graph helps too!!!

  3. If “it” goes in evenly, it’s a

  4. If “it” goes in evenly, it’s a FACTOR

  5. If “it” goes in evenly, it’s a FACTOR We know the remainder is 0

  6. Ultimate Goal • To solve polynomials • To graph polynomials • If we can get the polynomial in factored form, then life is pretty easy!

  7. Tool to help find factors • Long Division • Always works • Synthetic Division • Works with linear factors • Especially when written in the form (x-a)

  8. Tool to help find factors • We need a BANK to limit the choices • RATIONAL ROOT THEOREM • If your polynomial is in standard form and • If your polynomial has integer coefficients • Then the bank consists of • The possible rational roots are found in the bank

  9. Example • List the possible rational roots for

  10. Before we extend this…

  11. BOGO, new perspective • Irrational Root Theorem • If the is a root, then is also a root • If is a root, then is also a root • Imaginary Root Theorem • If is a root, then is also a root • They are called conjugates of each other!

  12. Examples • Given the polynomial, find the other roots

  13. Intermission • Stand up and stretch for 1 minute • Then use your calc to answer pg 318 #49

  14. Back to finding all the roots • Solve:

  15. Back to finding all the roots • Solve: • Solve:

  16. Back to finding all the roots • Solve: • Solve: • Solve:

  17. Fundamental Theorem of Algebra • (Corollary) nth degree polynomials have exactly n roots (including real and imaginary) • has 3 roots. Period. • has 4 roots. Period.

  18. All done! • Solve

  19. All done! • Solve • Solve

  20. Study • Form a study group tonight! • Do 6.5 pg 339 #4, 7, 12, 13, 17, 19, 21, 38 • Do 6.6 pg 343 #10-12 • Download Geist’s notes (from my website) • Review Solving Quadratics • Do Chap 5 Review pg. 300 #24, 27, 31, 38 • Friday test: Chap 6 / part of Chap 5 Retest • Monday: Review for Cumulative • Tuesday: Cumulative 5-6 test • Wed: Prepare for Final and take probability test • Friday: 200 point Final

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