Algebra: The Polynomial Arithmetic
The following sets of problems are about:
- Addition of Polynomials
- Subtraction of Polynomials
- Multiplication of Polynomials
- Division of Polynomials
Addition of Polynomials
| 1. | 9x - y - 2z and 7x + 5y - 8z |
| Answer: 16x + 4y - 10z |
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| 2. | 8m - 7n + 7 and 5m + 3n - 2 |
| Answer: 13m - 4n + 5 |
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| 3. | 5a + 3b + 9 and 2a - 9b + 7 |
| Answer: 7a - 6b + 16 |
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| 4. | 9a + 4b + 9c and 3a + 3b + 2c |
| Answer: 12a + 7b + 11c |
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| 5. | 6x - 7y - 6z, 9x + 8y - 7z and 9x - 5y - 7z |
| Answer: 24x - 4y - 20z |
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| 6. | 2x - 9y + 2z, 6x - 5y + z and 7x + 6y + 9 |
| Answer: 15x - 8y + 3z + 9 |
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| 7. | 5a + 6b - 3, 1a + b - 6c and 3a - 7b + 5 |
| Answer: 9a - 6c + 2 |
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| 8. | 6x - 7y - 6z, 9x + 8y - 7z and 9x - 5y - 7z |
| Answer: 24x - 4y - 20z |
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| 9. | 2x - 9y + 2z, 6x - 5y + z and 7x + 6y + 9 |
| Answer: 15x - 8y + 3z + 9 |
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| 10. | 3a + 5b + 9, 8a - 9b - 7c and 9a + b - 3 |
| Answer: 20a - 3b - 7c + 6 |
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| 11. | 6a2 - 6ab - 12a and 7a2 - 28ab - 7a |
| Answer: 13a2 - 34ab - 19a |
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| 12. | 14a2b + 56ab2 + 21ab and 42a2b - 7ab2 - 7ab |
| Answer: 56a2b + 49ab2 + 14ab |
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| 13. | 3a5 + 5a4b + 8a4 and 20a5 + 35a4b - 10a4 |
| Answer: 23a5 + 40a4b - 2a4 |
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| 14. | 35x2 + 28xy + 21x and 20x2 + 4xy - 28x |
| Answer: 55x2 + 32xy - 7x |
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| 15. | 10x2y + 5xy2 + 40xy and 49x2y + 14xy2 + 7xy |
| Answer: 59x2y + 19xy2 + 47xy |
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| 16. | 49x5 - 35x4y + 49x4 and 8x5 + 14x4y + 14x4 |
| Answer: 57x5 - 21x4y + 63x4 |
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| 17. | 4x4 + x3 + 9 and 2x4 + 7x3 + 3 |
| Answer: 6x4 + 8x3 + 12 |
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| 18. | 8m3 + m2 - 6 and 7m3 + 8m2 - 7 |
| Answer: 15m3 + 9m2 - 13 |
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| 19. | 4b2 + 9b + 5 and 2b2 + 5b + 2 |
| Answer: 6b2 + 14b + 7 |
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| 20. | 63x2 - 9xy - 63x and 30x2 - 40xy - 45x |
| Answer: 93x2 - 49xy - 108x |
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Subtraction of Polynomials
| 1. | 9x + 2y + 3z and 7x + 6y + 8z |
| Answer: 2x - 4y - 5z |
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| 2. | 2m + 8n + 2 and m - 7n - 8 |
| Answer: m + 15n + 10 |
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| 3. | 8a + 2b - 1 and 7a - b - 3 |
| Answer: a + 3b + 2 |
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| 4. | 6a + 5b + 8c and 4a - 7b + 7c |
| Answer: 2a + 12b + c |
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| 5. | 6x - 3y + 8z, x - 3y - 5z and 2x + 7y - 2z |
| Answer: 3x - 7y + 15z |
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| 6. | 9x + 4y + 9z, 4x - 7y + z and x + 5y - 2 |
| Answer: 4x + 6y + 8z + 2 |
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| 7. | 9a + 3b - 5, 8a + 7b + 4c and a - b + 4 |
| Answer: - 3b - 4c - 9 |
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| 8. | 8x - 4y + 5z, 3x + 4y - 7z and 5x + 4y - z |
| Answer: -12y + 13z |
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| 9. | 9x - 5y - 9z, 2x + 2y - 3z and 5x + 6y - 5 |
| Answer: 2x - 13y - 6z + 5 |
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| 10. | 9a + 3b - 8, a + 5b + 2c and 7a - 8b + 3 |
| Answer: a + 6b - 2c - 11 |
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| 11. | 4a2 - 20ab - 4a and a2 - 8ab + 3a |
| Answer: 3a2 - 12ab - 7a |
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| 12. | 63a2b + 36ab2 - 45ab and 9a2b - ab2 - 5ab |
| Answer: 54a2b + 37ab2 - 40ab |
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| 13. | 24a5 - 21a4b - 24a4 and 20a5 + 35a4b - 45a4 |
| Answer: 4a5 - 56a4b + 21a4 |
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| 14. | 24x2 + 24xy + 9x and 12x2 + 24xy - 4x |
| Answer: 12x2 + 13x |
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| 15. | 21x2y + 42xy2 + 7xy and 9x2y + 9xy2 + 63xy |
| Answer: 12x2y + 33xy2 - 56xy |
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| 16. | 18x3 - 63x2y - 81x2 and 6x3 - 5x2y - 8x2 |
| Answer: 12x3 - 58x2y - 73x2 |
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| 17. | 9x2 + 2x + 7 and 4x2 + 3x + 8 |
| Answer: 5x2 - x - 1 |
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| 18. | 9m2 - 7m + 6 and 8m2 - 2m + 1 |
| Answer: m2 - 5m + 5 |
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| 19. | 8b2 - 3b + 1 and 7b2 - b + 5 |
| Answer: b2 - 2b - 4 |
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| 20. | 5x2 - 7x + 8 and 4x2 - 5x + 7 |
| Answer: x2 - 2x + 1 |
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Multiplication of Polynomials
| 1. | x - 3y and x - 7y |
| Answer: x2 - 10xy + 21y2 |
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| 2. | 4a - 7b and 4a - 7b |
| Answer: 16a2 - 56ab + 49b2 |
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| 3. | 6a + b and 6a + b |
| Answer: 36a2 + 12ab + b2 |
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| 4. | 8x + y and 9x + 4y |
| Answer: 72x2 + 41xy + 4y2 |
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| 5. | 2x - 7y and 2x + 7y |
| Answer: 4x2 - 49y2 |
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| 6. | 3a - 4b and 9a + 2b |
| Answer: 27a2 - 30ab - 8b2 |
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| 7. | 6mx + ny and 8mx - ny |
| Answer: 48m2x2 + 2mxny - n2y2 |
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| 8. | 7ax + by and 8ax + 7by |
| Answer: 56a2x2 + 57axby + 7b2y2 |
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| 9. | (x - 5y - 6)2 |
| Answer: x2 - 10xy + 25y2 - 12x + 60y + 36 |
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| 10. | (8p - 7q + 1)2 |
| Answer: 64p2 - 112pq + 49q2 + 16p - 14q + 1 |
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| 11. | (8p - 7q + 4)2 |
| Answer: 64p2 - 112pq + 49q2 + 64p - 56q + 16 |
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| 12. | (a - 5b - 6y)2 |
| Answer: a2 - 10ab + 25b2 - 12a + 60b + 36y2 |
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| 13. | (5x4)(x + y) |
| Answer: 5x5 + 5x4y |
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| 14. | (4x)(7x4 - 8y2) |
| Answer: 28x5 - 32xy2 |
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| 15. | (20xy)(3x4 - 8) |
| Answer: 60x5y - 160xy |
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| 16. | (3ab4)(3a4 + 7) |
| Answer: 9a5b4 + 21ab4 |
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| 17. | (14s3t2)(3s3 - 7u2 + 7) |
| Answer: 42s6t2 - 98s3t2u2 + 98s3t2 |
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| 18. | (s + t)(u3 + v4) |
| Answer: su3 + sv4 + tu3 + tv4 |
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| 19. | (w + 4x)(5y + z) |
| Answer: 5wy + wz + 20xy + 4xz |
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| 20. | (5s + 7t)(7u - 4v) |
| Answer: 35su - 20sv + 49tu - 28tv |
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| 21. | (3w3 + 8x4)(2y2 + z3) |
| Answer: 6w3y2 + 3w3z3 + 16x4y2 + 8x4z3 |
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| 22. | (3w + 4x2)(4y2 - 3z4) |
| Answer: 12wy2 - 9wz4 + 16x2y2 - 12x2z4 |
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| 23. | (7a + 9b)(2b + 3) |
| Answer: 14ab + 21a + 18b2 + 27b |
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| 24. | (x + 9y)(7y - 4) |
| Answer: 7xy - 4x + 63y2 - 36y |
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| 25. | (8s3 + t)(4t4 - 7) |
| Answer: 32s3t4 - 56s3 + 4t5 - 7t |
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| 26. | (s - t)(u + 8v) |
| Answer: su + 8sv - tu - 8tv |
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| 27. | (x + 6)(x + 7) |
| Answer: x2 + 13x + 42 |
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| 28. | (x - 7)(x + 4) |
| Answer: x2 - 3x - 28 |
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| 29. | (x - 9)(x - 1) |
| Answer: x2 - 10x + 9 |
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| 30. | (8x + 1)(3x + 2) |
| Answer: 24x2 + 19x + 2 |
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| 31. | (x - 5)(4x + 3) |
| Answer: 4x2 - 17x - 15 |
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| 32. | (9x - 1)(5x - 2) |
| Answer: 45x2 - 23x + 2 |
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| 33. | (x + 9)(x + 9) |
| Answer: x2 + 18x + 81 |
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| 34. | (3x + 5)(3x + 5) |
| Answer: 9x2 + 30x + 25 |
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| 35. | (x - 1)(x - 1) |
| Answer: x2 - 2x + 1 |
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| 36. | (9x - 1)(9x - 1) |
| Answer: 81x2 - 18x + 1 |
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| 37. | [ 9x + 8(y - z) ] [ (9x - 8(y - z) ] |
| Answer: 81x2 - 64(y - z)2 |
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| 38. | (x + 2y) (x - 2y) (x2 + 4y2) |
| Answer: x4 - 16y4 |
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| 39. | (x - 8) (x2 + 8x + 64) |
| Answer: x3 - 512 |
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| 40. | (x2 - 7) (x4 + 7x2 + 49) |
| Answer: x6 - 343 |
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| 41. | (7y - 8) (49y2 + 56y + 64) |
| Answer: 343y3 - 512 |
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| 42. | (2y2 - 9) (4y4 + 18y2 + 81) |
| Answer: 8y6 - 729 |
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| 43. | (3a - 4b) (9a2 + 12ab + 16b2) |
| Answer: 27a3 - 64b3 |
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| 44. | (3a2 - 7b) (9a4 + 21a2b + 49b2) |
| Answer: 27a6 - 343b3 |
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| 45. | (9xy2 - 4z) (81x2y4 + 36xy2z + 16z2) |
| Answer: 729x3y6 - 64z3 |
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| 46. | x (2s + 9t)(4s2 - 18st + 81t2) |
| Answer: 8s3x + 729t3x |
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| 47. | x (8s2 + 2t2)(64s4 - 16s2t2 + 4t4) |
| Answer: 512s6x + 8t6x |
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| 48. | (8x + 5y) (64x2 - 40xy + 25y2) |
| Answer: 512x3 + 125y3 |
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| 49. | (7x2 + 9y2) (49x4 - 63x2y2 + 81y4) |
| Answer: 343x6 + 729y6 |
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| 50. | 4x9 (y + 7z) (y - 7z) |
| Answer: 4x9y2 - 196x9z2 |
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Division of Polynomials
| 1. | 63a4b7x9 ÷ 9a3b5x4 |
| Answer: 7ab2x5 |
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| 2. | 9x7y6z4 ÷ x2y3z3 |
| Answer: 9x5y3z |
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| 3. | -14m4n7 ÷ 2m3n4 |
| Answer: -7mn3 |
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| 4. | 72x2y2 ÷ 8xy |
| Answer: 9xy |
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| 5. | 24a6b5c7 ÷ 3a2b4c3 |
| Answer: 8a4bc4 |
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| 6. | 28x5y2z + 63x3y3z + 49x2y3z ÷ 7x2y2z |
| Answer: 4x3 + 9xy + 7y |
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| 7. | x2 - 8x - 209 ÷ x - 19 |
| Answer: x + 11 |
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| 8. | x2 - 29x + 204 ÷ x - 9 |
| Answer: x - 20; remainder = 24 |
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| 9. | x2 - 22x + 88 ÷ x - 4 |
| Answer: x - 18; remainder = 16 |
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| 10. | x4 + 3x2 - 460 ÷ x2 + 23 |
| Answer: x2 - 20 |
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| 11. | x4 - 21x2 + 130 ÷ x2 - 10 |
| Answer: x2 - 11; remainder = 20 |
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| 12. | x4 - 6x3y - 25x2y2 + 52xy3 + 14y4 ÷ x2 + 2xy - 7y2 |
| Answer: x2 - 8xy - 2y2 |
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| 13. | x4 - 8x3y + 7x2y2 + 34xy3 + 15y4 ÷ x2 - 5xy - 3y2 |
| Answer: x2 - 3xy - 5y2 |
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| 14. | 8x3 - 57x2 + 48x + 42 ÷ x - 6 |
| Answer: 8x2 - 9x - 6; remainder = 6 |
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| 15. | 56x5 - 111x4 + 94x3 - 70x2 + 27x - 47 ÷ 7x3 - 6x2 - 2x - 5 |
| Answer: 8x2 - 9x + 8; remainder = -2x - 7 |
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| 16. | 6a5x4y9 ÷ a3x6y6 |
| Answer: 6a2x-2y3 |
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| 17. | 56p9q2r4 ÷ 8p3q6r5 |
| Answer: 7p6q-4r-1 |
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| 18. | 63x5 + 88x4 + 41x3 - 63x - 63x2 + 9 ÷ 7x3 + 9x2 + 9x - 1 |
| Answer: 9x2 + x - 7; remainder = x + 2 |
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| 19. | 32x3 + 44x2 + 25x + 21 ÷ 8x + 9 |
| Answer: 4x2 + x + 2; remainder = 3 |
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| 20. | x4 + 11x3y + 25x2y2 - 7xy3 - 6y4 ÷ x2 + 8xy + 3y2 |
| Answer: x2 + 3xy - 2y2 |
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