1 / 9

Polynomials and roots

Polynomials and roots. Jeffrey Bivin Lake Zurich High School Jeff.bivin@lz95.org. Last Updated: October 26, 2009. Write a 4 th degree polynomial with the given the roots of 1, 2, 3, 4. F(x) = (x – 1)(x – 2)(x – 3)(x – 4). F(x) = (x 2 – 3x + 2)(x 2 – 7x + 12).

Download Presentation

Polynomials and roots

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. Polynomials and roots Jeffrey Bivin Lake Zurich High School Jeff.bivin@lz95.org Last Updated: October 26, 2009

  2. Write a 4th degree polynomial with the given the roots of 1, 2, 3, 4 F(x) = (x – 1)(x – 2)(x – 3)(x – 4) F(x) = (x2 – 3x + 2)(x2 – 7x + 12) F(x) = x4 – 7x3 + 12x2 -3x3 + 21x2 - 36x 2x2 - 14x + 24 F(x) = x4 – 10x3 + 35x2 – 50x + 24

  3. Given the 4 numbers 1, 2, 3, 4 Find the product of the four numbers: 1•2•3•4 = 24 Find all groups of three of the four numbers and find each product: 1•2•3 = 6 1•2•4 = 8 1•3•4 = 12 2•3•4 = 24 Now add their products: 6 + 8 + 12 + 24 = 50 Find all groups of two of the four numbers and find each product: 1•2 = 2 1•3 = 3 1•4 = 4 2•3 = 6 2•4 = 8 3•4 = 12 Now add their products: 2 + 3 + 4 + 6 + 8 + 12 = 35 Find all groups of one of the four numbers and find each product: Now add their products: 1+ 2 + 3 + 4 = 10

  4. Write a 4th degree polynomial with the given the roots of 1, 2, 3, 4 F(x) = (x – 1)(x – 2)(x – 3)(x – 4) F(x) = (x2 – 3x + 2)(x2 – 7x + 12) F(x) = x4 – 7x3 + 12x2 -3x3 + 21x2 - 36x 2x2 - 14x + 24 opposite opposite same same F(x) = x4 – 10x3 + 35x2 – 50x + 24

  5. Write a 5th degree polynomial with the given the roots of 5, 1, 2, 3, 4 F(x) = (x - 5)(x – 1)(x – 2)(x – 3)(x – 4) F(x) = (x – 5)(x4 – 10x3 + 35x2 – 50x + 24) F(x) = x5 – 10x4 + 35x3 – 50x2 + 24x -5x4 + 50x3 – 175x2 + 250x – 120 F(x) = x5 – 15x4 + 85x3 – 225x2 + 274x - 120

  6. Given the 5 numbers 5, 1, 2, 3, 4 Find the product of the five numbers: 5•1•2•3•4 = 120 Find all groups of four of the five numbers and find each product: 2•3•4•5 = 120 1•2•3•4 = 24 1•2•3•5 = 30 1•2•4•5 = 40 1•3•4•5 = 60 Now add: 24 + 30 + 40 + 60 + 120 = 274 Find all groups of three of the five numbers and find each product: 1•2•3 = 6 1•2•4 = 8 1•2•5 = 10 1•3•4 = 12 1•3•5 = 15 1•4•5 = 20 2•3•4 = 24 2•3•5 = 30 2•4•5 = 40 3•4•5 = 60 Now add: 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 = 225 Find all groups of two of the five numbers and find each product: 1•2 = 2 1•3 = 3 1•4 = 4 1•5 = 5 2•3 = 6 2•4 = 8 2•5 = 10 3•4 = 12 3•5 = 15 4•5 = 20 Now add: 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 +15 + 20 = 85 Find all groups of one of the five numbers and find each product: Now add: 1 + 2 + 3 + 4 + 5 = 15

  7. Write a 5th degree polynomial with the given the roots of 5, 1, 2, 3, 4 opposite 15 same 85 opposite 225 same 274 opposite 120 F(x) = x5 – 15x4 + 85x3 – 225x2 + 274x – 120

  8. Given the 5 numbers 3, 1±2i Find the product of the three numbers: opposite 3(1+2i)(1-2i) = 3(1 - 4i2) = 3(1 + 4) = 3(5) =15 Find all groups of two of the five numbers and find each product: 3•(1 + 2i) = 3 + 6i 3•(1 – 2i) = 3 – 6i (1 + 2i)(1 – 2i) = 5 same Now add: 3 + 6i + 3 – 6i + 5 = 11 Find all groups of one of the five numbers and find each product: opposite Now add: 3 + 1 + 2i + 1 – 2i = 5 F(x) = x3 – 5x2 + 11x – 15

  9. Write a 3rd degree polynomial with the given the roots of 3, 1±2i F(x) = (x – 3)(x – (1+2i))(x – (1–2i)) F(x) = (x – 3)(x – 1 – 2i)(x – 1 + 2i) F(x) = (x – 3)((x – 1) – 2i)((x – 1) + 2i) F(x) = (x – 3)((x – 1)2 – 4i2) F(x) = (x – 3)(x2 – 2x + 1 + 4) F(x) = (x – 3)(x2 – 2x + 5) F(x) = x3 – 2x2 + 5x – 3x2 + 6x – 15 F(x) = x3 – 5x2 + 11x – 15

More Related