1 / 27

2.3 Polynomial and Synthetic Division

2.3 Polynomial and Synthetic Division. Why teach long division in grade school?. Long Division. Find 2359 ÷ 51 by hand Which one goes inside √. Long Division. Find 2359 ÷ 51 by hand 51 √ 2359. Long Division. Find 2359 ÷ 51 by hand 4 51 √ 2359 204

andromeda-marinos
Download Presentation

2.3 Polynomial and Synthetic Division

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. 2.3 Polynomial and Synthetic Division Why teach long division in grade school?

  2. Long Division Find 2359 ÷ 51 by hand Which one goes inside √

  3. Long Division Find 2359 ÷ 51 by hand 51 √ 2359

  4. Long Division Find 2359 ÷ 51 by hand 4 51 √ 2359 204 What do you do now ?

  5. Long Division Find 2359 ÷ 51 by hand 4 51 √ 2359 - 204 319

  6. Long Division Find 2359 ÷ 51 by hand 46 51 √ 2359 - 204 319 - 306 13

  7. Long Division Find 2359 ÷ 51 by hand 46 13/51 51 √ 2359 - 204 319 - 306 13

  8. Lets do the same with a Polynomial Divide 6x3 + 4x2 – 10x – 5 by 2x2 + 1 2x2 + 1 √ 6x3 + 4x2 – 10x - 5

  9. Lets do the same with a Polynomial Divide 6x3 + 4x2 – 10x – 5 by 2x2 + 1 3x 2x2 + 1 √ 6x3 + 4x2 – 10x – 5 6x3 + 3x

  10. Lets do the same with a Polynomial Divide 6x3 + 4x2 – 10x – 5 by 2x2 + 1 3x 2x2 + 1 √ 6x3 + 4x2 – 10x – 5 6x3 + 3x 4x2 – 13x - 5

  11. Lets do the same with a Polynomial Divide 6x3 + 4x2 – 10x – 5 by 2x2 + 1 3x + 2 2x2 + 1 √ 6x3 + 4x2 – 10x – 5 6x3 + 3x 4x2 – 13x - 5 4x2 + 2 - 13x - 7

  12. Lets do the same with a Polynomial Divide 6x3 + 4x2 – 10x – 5 by 2x2 + 1 3x + 2 + 2x2 + 1 √ 6x3 + 4x2 – 10x – 5 6x3 + 3x 4x2 – 13x - 5 4x2 + 2 - 13x - 7

  13. The Division Algorithm f(x) = d(x)g(x) + r(x) (6x3 + 4x2 – 10x – 5) = (3x + 2)(2x2 + 1) +(-13x – 7) = 6x3 + 4x2 + 3x + 2 – 13x – 7 = 6x3 + 4x2 - 10x – 5 WE can use the division Algorithm to find G.C.D. (greatest common divisors )

  14. What is the G.C.D. of 3461, 4879 4879 = 3461(1) + 1418 3461 = 1418(2) + 625 1418 = 625(2) + 168 625 = 168(3) + 121 168 = 121(1) + 47 121 = 47(2) + 27 47 = 27(1) + 20 27 = 20(1) + 7 20 = 7(2) + 6 7 = 6(1) + 1 ← G.C.D. 6 = 1(6) + 0

  15. Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italianmathematician and philosopher. By 1788 he had earned university degrees in philosophy, medicine/surgery, and mathematics. Among his work was an incomplete proof (Abel–Ruffini theorem1) that quintic (and higher-order) equations cannot be solved by radicals (1799), and Ruffini's rule3 which is a quick method for polynomial division. Ruffini’s rule3 or Synthetic Division

  16. Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 x = 2 This will go in the little box in the first line.

  17. Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 2 | 4 5 0 8 The coefficients are written out in descending exponential order. (even leaving a zero for the 1st degree term)

  18. Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 2 | 4 5 0 8 4 The first number is dropped, then multiply by 2 and add to 5

  19. Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 2 | 4 5 0 8 8 4 13 Then the steps are repeated added and multiply by 2.

  20. Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 2 | 4 5 0 8 8 26 52 4 13 26 60 Then the steps are repeated added and multiply by 2.

  21. Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 2 | 4 5 0 8 8 26 52 4 13 26 60 60 is the reminder; 26 is the constant, 13 the 1st degree term, 4 the 2nd degree term

  22. Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 2 | 4 5 0 8 8 26 52 4 13 26 60 4x2 + 13x + 26 +

  23. The Remainder Theorem The remainder is the answer! So in f(x) = 4x3 + 5x2 + 8 f(2) = 60

  24. The Remainder Theorem The remainder is the answer! So in f(x) = 4x3 + 5x2 + 8 f(2) = 60 Check it out: 4(2)3 + 5(2)2 + 8 4(8) + 5(4) + 8 32 + 20 + 8 = 60

  25. (x2 + 3x – 40) ÷ (x - 5) 5| 1 3 - 40 5 40 1 8 0 Since the reminder is 0, 5 is a root or zero of the equation. What is the other root?

  26. Homework Page 140 – 142 # 2, 8, 14, 17, 21, 24, 28, 36, 42, 44, 51, 55, 63, 74, 82, 92

  27. Homework Page 140-142 # 7, 13, 15, 22, 27, 35, 40, 43, 50, 53, 62, 70, 81, 86

More Related