4 7polynomial operations-vertical
- 2. Division of Polynomials In arithmetic we +, – , and * whole numbers and obtaining whole number answers.
- 3. Division of Polynomials In arithmetic we +, – , and * whole numbers and obtaining whole number answers. Likewise in algebra, we +, – , and * polynomials and obtaining polynomial answers.
- 4. Division of Polynomials In arithmetic we +, – , and * whole numbers and obtaining whole number answers. Similarly there is a long division algorithm for polynomials that corresponds to dividing numbers. Likewise in algebra, we +, – , and * polynomials and obtaining polynomial answers.
- 5. Division of Polynomials In arithmetic we +, – , and * whole numbers and obtaining whole number answers. Similarly there is a long division algorithm for polynomials that corresponds to dividing numbers. Just as the case for dividing numbers, we will do the case where the division yields no remainder. Likewise in algebra, we +, – , and * polynomials and obtaining polynomial answers.
- 6. Division of Polynomials In arithmetic we +, – , and * whole numbers and obtaining whole number answers. Similarly there is a long division algorithm for polynomials that corresponds to dividing numbers. Just as the case for dividing numbers, we will do the case where the division yields no remainder. We will come back for the case that yields a remainder after we learn how to do fractional algebra so we may check our answers. Likewise in algebra, we +, – , and * polynomials and obtaining polynomial answers.
- 7. Division of Polynomials In arithmetic we +, – , and * whole numbers and obtaining whole number answers. Similarly there is a long division algorithm for polynomials that corresponds to dividing numbers. Just as the case for dividing numbers, we will do the case where the division yields no remainder. We will come back for the case that yields a remainder after we learn how to do fractional algebra so we may check our answers. We start by reviewing the long division for the whole numbers. Likewise in algebra, we +, – , and * polynomials and obtaining polynomial answers.
- 8. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 Division
- 9. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 Division Steps. i. (Front-in Back-out) Put the problem in a scaffold format with the front-number, the dividend, inside the scaffold, and the back-number, the divisor, outside the scaffold.
- 10. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 Steps. i. (Front-in Back-out) Put the problem in a scaffold format with the front-number, the dividend, inside the scaffold, and the back-number, the divisor, outside the scaffold.“the back” outside 2 78 “the front” inside Division
- 11. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 Steps. i. (Front-in Back-out) Put the problem in a scaffold format with the front-number, the dividend, inside the scaffold, and the back-number, the divisor, outside the scaffold.“the back” outside 2 78 “the front” inside Division ii. Enter the quotient on top, multiply the quotient back into the scaffold and subtract the results from the dividend, bring down the unused digits, if any. This is the new dividend.
- 12. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 Steps. i. (Front-in Back-out) Put the problem in a scaffold format with the front-number, the dividend, inside the scaffold, and the back-number, the divisor, outside the scaffold.“the back” outside 2 78 “the front” inside Enter the quotient on top if possible 3 Division ii. Enter the quotient on top, multiply the quotient back into the scaffold and subtract the results from the dividend, bring down the unused digits, if any. This is the new dividend.
- 13. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 ii. Enter the quotient on top, multiply the quotient back into the scaffold and subtract the results from the dividend, bring down the unused digits, if any. This is the new dividend. Steps. i. (Front-in Back-out) Put the problem in a scaffold format with the front-number, the dividend, inside the scaffold, and the back-number, the divisor, outside the scaffold.“the back” outside 2 78 “the front” inside Enter the quotient on top if possible 3 62 x 3 18 Division multiply the quotient back into the scaffold, subtract
- 14. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 ii. Enter the quotient on top, multiply the quotient back into the scaffold and subtract the results from the dividend, bring down the unused digits, if any. This is the new dividend. Steps. i. (Front-in Back-out) Put the problem in a scaffold format with the front-number, the dividend, inside the scaffold, and the back-number, the divisor, outside the scaffold.“the back” outside 2 78 “the front” inside Enter the quotient on top if possible 3 62 x 3 18 Division new dividendmultiply the quotient back into the scaffold, subtract
- 15. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 ii. Enter the quotient on top, multiply the quotient back into the scaffold and subtract the results from the dividend, bring down the unused digits, if any. This is the new dividend. Enter a quotient and repeat step ii . Steps. i. (Front-in Back-out) Put the problem in a scaffold format with the front-number, the dividend, inside the scaffold, and the back-number, the divisor, outside the scaffold.“the back” outside 2 78 “the front” inside Enter the quotient on top if possible 3 62 x 3 18 Division new dividendmultiply the quotient back into the scaffold, subtract
- 16. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 ii. Enter the quotient on top, multiply the quotient back into the scaffold and subtract the results from the dividend, bring down the unused digits, if any. This is the new dividend. Enter a quotient and repeat step ii . Steps. i. (Front-in Back-out) Put the problem in a scaffold format with the front-number, the dividend, inside the scaffold, and the back-number, the divisor, outside the scaffold.“the back” outside 2 78 “the front” inside Enter the quotient on top if possible 39 62 x 3 18 Division new dividendmultiply the quotient back into the scaffold, subtract
- 17. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 ii. Enter the quotient on top, multiply the quotient back into the scaffold and subtract the results from the dividend, bring down the unused digits, if any. This is the new dividend. Enter a quotient and repeat step ii . Steps. i. (Front-in Back-out) Put the problem in a scaffold format with the front-number, the dividend, inside the scaffold, and the back-number, the divisor, outside the scaffold.“the back” outside 2 78 “the front” inside Enter the quotient on top if possible 39 62 x 3 18 Division new dividend 18 02 x 9 multiply the quotient back into the scaffold, subtract
- 18. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 ii. Enter the quotient on top, multiply the quotient back into the scaffold and subtract the results from the dividend, bring down the unused digits, if any. This is the new dividend. Enter a quotient and repeat step ii . Steps. i. (Front-in Back-out) Put the problem in a scaffold format with the front-number, the dividend, inside the scaffold, and the back-number, the divisor, outside the scaffold.“the back” outside 2 78 “the front” inside Enter the quotient on top if possible 39 iii. If the new dividend is not enough to be divided by the divisor, STOP! This is the remainder R. 62 x 3 18 Division stop new dividend 18 02 x 9 multiply the quotient back into the scaffold, subtract
- 19. We demonstrate the vertical long-division of numbers below. The Vertical Format Example A. Divide 78 ÷ 2 ii. Enter the quotient on top, multiply the quotient back into the scaffold and subtract the results from the dividend, bring down the unused digits, if any. This is the new dividend. Enter a quotient and repeat step ii . Steps. i. (Front-in Back-out) Put the problem in a scaffold format with the front-number, the dividend, inside the scaffold, and the back-number, the divisor, outside the scaffold.“the back” outside 2 78 “the front” inside Enter the quotient on top if possible 39 iii. If the new dividend is not enough to be divided by the divisor, STOP! This is the remainder R. 62 x 3 18 So the remainder R is 0 and we have that 78 ÷ 2 = 39 evenly and that 2 x 39 = 78. Division stop new dividend 18 02 x 9 multiply the quotient back into the scaffold, subtract
- 20. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 D(x) N(x)Numerator Denominator
- 21. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 Set up for the division N(x) ÷ D(x) the same way for dividing numbers.D(x) N(x)Numerator Denominator
- 22. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 1. Set up the long division, N(x) inside, D(x) outside. 2x2 – 3x + 20x – 4 Set up for the division N(x) ÷ D(x) the same way for dividing numbers.D(x) N(x)Numerator Denominator
- 23. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 1. Set up the long division, N(x) inside, D(x) outside. 2x2 – 3x + 20x – 4 2x Set up for the division N(x) ÷ D(x) the same way for dividing numbers.D(x) N(x)Numerator Denominator 2. Enter on top the quotient of the leading terms .
- 24. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 1. Set up the long division, N(x) inside, D(x) outside. 2x2 – 3x + 20x – 4 2x 2x2 – 8x 3. Multiply this quotient to the divisor and subtract the result from the dividend. Set up for the division N(x) ÷ D(x) the same way for dividing numbers.D(x) N(x)Numerator Denominator 2. Enter on top the quotient of the leading terms .
- 25. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 1. Set up the long division, N(x) inside, D(x) outside. 2x2 – 3x + 20x – 4 2x 2x2 – 8x 3. Multiply this quotient to the divisor and subtract the result from the dividend. – ) – + Set up for the division N(x) ÷ D(x) the same way for dividing numbers.D(x) N(x)Numerator Denominator 2. Enter on top the quotient of the leading terms .
- 26. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 1. Set up the long division, N(x) inside, D(x) outside. 2x2 – 3x + 20x – 4 2x 2x2 – 8x 3. Multiply this quotient to the divisor and subtract the result from the dividend. – ) – + 5x + 20 Set up for the division N(x) ÷ D(x) the same way for dividing numbers.D(x) N(x)Numerator Denominator 2. Enter on top the quotient of the leading terms .
- 27. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 1. Set up the long division, N(x) inside, D(x) outside. 2x2 – 3x + 20x – 4 2x 2x2 – 8x 3. Multiply this quotient to the divisor and subtract the result from the dividend. – ) – + 5x + 20 Set up for the division N(x) ÷ D(x) the same way for dividing numbers.D(x) N(x)Numerator Denominator new dividend 2. Enter on top the quotient of the leading terms . 4. This difference is the new dividend, repeat steps 2 and 3.
- 28. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 1. Set up the long division, N(x) inside, D(x) outside. 2x2 – 3x + 20x – 4 2x + 5 2x2 – 8x 3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. This difference is the new dividend, repeat steps 2 and 3. – ) – + 5x + 20 Set up for the division N(x) ÷ D(x) the same way for dividing numbers.D(x) N(x)Numerator Denominator new dividend 2. Enter on top the quotient of the leading terms .
- 29. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 1. Set up the long division, N(x) inside, D(x) outside. 2x2 – 3x + 20x – 4 2x + 5 2x2 – 8x 3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. This difference is the new dividend, repeat steps 2 and 3. – ) – + 5x + 20 5x – 20– ) – + 0 Set up for the division N(x) ÷ D(x) the same way for dividing numbers.D(x) N(x)Numerator Denominator new dividend 2. Enter on top the quotient of the leading terms . the remainder
- 30. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 1. Set up the long division, N(x) inside, D(x) outside. 2x2 – 3x + 20x – 4 2x + 5 2x2 – 8x 3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. This difference is the new dividend, repeat steps 2 and 3. 5. Stop when the degree of the new dividend is smaller than the degree of the divisor, i.e. no more quotient is possible. – ) – + 5x + 20 5x – 20– ) – + 0 Set up for the division N(x) ÷ D(x) the same way for dividing numbers.D(x) N(x)Numerator Denominator new dividend 2. Enter on top the quotient of the leading terms . the quotient Q(x) the remainder
- 31. The Long Division Example B. Divide using long division.x – 4 2x2 – 3x + 20 1. Set up the long division, N(x) inside, D(x) outside. 2x2 – 3x + 20x – 4 2x + 5 2x2 – 8x 3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. This difference is the new dividend, repeat steps 2 and 3. 5. Stop when the degree of the new dividend is smaller than the degree of the divisor, i.e. no more quotient is possible. – ) – + 5x + 20 5x – 20– ) – + 0 Set up for the division N(x) ÷ D(x) the same way for dividing numbers.D(x) N(x)Numerator Denominator new dividend 2. Enter on top the quotient of the leading terms . The remainder is 0, so that (x – 4)(2x + 5) = 2x2 – 3x + 20 the quotient Q(x) the remainder
- 32. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). D(x) N(x)
- 33. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 D(x) N(x)
- 34. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 D(x) N(x)
- 35. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x D(x) N(x)
- 36. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x x3 + x D(x) N(x)
- 37. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x x3 + x– ) –– D(x) N(x)
- 38. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x x3 + x– ) –– – 2x2 – 2 D(x) N(x)
- 39. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x – 2 x3 + x– ) –– – 2x2 – 2 D(x) N(x)
- 40. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x – 2 x3 + x– ) –– – 2x2 – 2 – 2x2 – 2 D(x) N(x)
- 41. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x – 2 x3 + x– ) – – ) ++ – – 2x2 – 2 – 2x2 – 2 D(x) N(x)
- 42. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x – 2 x3 + x– ) – – ) ++ 0 – – 2x2 – 2 – 2x2 – 2 D(x) N(x) (the remainder)
- 43. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x – 2 x3 + x– ) – – ) ++ 0 – – 2x2 – 2 Hence x2 + 1 x3 – 2x2 + x – 2 = x – 2 – 2x2 – 2 D(x) N(x) (the remainder)
- 44. The Long Division (Long Division Theorem) If Q(x) is the quotient of with the remainder 0, then Q(x)D(x) = N(x). Example C. Divide using the long division x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x – 2 x3 + x– ) – – ) ++ 0 – – 2x2 – 2 Hence x2 + 1 x3 – 2x2 + x – 2 = x – 2 – 2x2 – 2 Check that: (x2 + 1)(x – 2) = x3 – 2x2 + x – 2 D(x) N(x) (the remainder)
- 45. The Long Division A variation on this method to embed the subtraction into the process by using the negative the divisor in the set up to avoid mistakes.
- 46. The Long Division A variation on this method to embed the subtraction into the process by using the negative the divisor in the set up to avoid mistakes. To divide using the long division,x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1
- 47. The Long Division A variation on this method to embed the subtraction into the process by using the negative the divisor in the set up to avoid mistakes. To divide using the long division, set up the division by changing x2 + 1 to – x2 – 1 to incorporate the subtraction operation. x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 – x2 – 1
- 48. The Long Division A variation on this method to embed the subtraction into the process by using the negative the divisor in the set up to avoid mistakes. To divide using the long division, set up the division by changing x2 + 1 to – x2 – 1 to incorporate the subtraction operation. x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x – x2 – 1
- 49. The Long Division A variation on this method to embed the subtraction into the process by using the negative the divisor in the set up to avoid mistakes. To divide using the long division, set up the division by changing x2 + 1 to – x2 – 1 to incorporate the subtraction operation. x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x –x3 – x – x2 – 1 + )
- 50. The Long Division A variation on this method to embed the subtraction into the process by using the negative the divisor in the set up to avoid mistakes. To divide using the long division, set up the division by changing x2 + 1 to – x2 – 1 to incorporate the subtraction operation. x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x –x3 – x 0 – 2x2 0 – 2 – x2 – 1 + )
- 51. The Long Division A variation on this method to embed the subtraction into the process by using the negative the divisor in the set up to avoid mistakes. To divide using the long division, set up the division by changing x2 + 1 to – x2 – 1 to incorporate the subtraction operation. x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x – 2 –x3 – x 0 – 2x2 0 – 2 – x2 – 1 + )
- 52. The Long Division A variation on this method to embed the subtraction into the process by using the negative the divisor in the set up to avoid mistakes. To divide using the long division, set up the division by changing x2 + 1 to – x2 – 1 to incorporate the subtraction operation. x2 + 1 x3 – 2x2 + x – 2 x3 – 2x2 + x – 2x2 + 1 x – 2 –x3 – x 0 0 – 2x2 0 – 2 + 2x2 + 2 – x2 – 1 + ) + )
- 53. The Long Division Need long division without remainder HW.