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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

vivian-kirkland
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2.3 Polynomial and Synthetic Division

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  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

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