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  • 1. Special Binomial Operations
  • 2. Special Binomial Operations A binomial is a two-term polynomial.
  • 3. Special Binomial Operations A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.
  • 4. Special Binomial Operations A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial.
  • 5. Special Binomial Operations A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c.
  • 6. Special Binomial Operations A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + #
  • 7. Special Binomial Operations A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # F: To get the x2-term, multiply the two Front x-terms of the binomials.
  • 8. Special Binomial Operations A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results.
  • 9. Special Binomial Operations A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results. L: To get the constant term, multiply the two Last constant terms.
  • 10. Special Binomial Operations A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results. L: To get the constant term, multiply the two Last constant terms. This is called the FOIL method.
  • 11. Special Binomial Operations A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results. L: To get the constant term, multiply the two Last constant terms. This is called the FOIL method. The FOIL method speeds up the multiplication of above binomial products and this will come in handy later.
  • 12. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4)
  • 13. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 The front terms: x2-term
  • 14. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 Outer pair: –4x
  • 15. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 Inner pair: –4x + 3x
  • 16. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x Outer Inner pairs: –4x + 3x = –x
  • 17. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
  • 18. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5)
  • 19. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 The front terms: –6x2
  • 20. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 Outer pair: 15x
  • 21. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 Inner pair: 15x – 8x
  • 22. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 + 7x Outer and Inner pair: 15x – 8x = 7x
  • 23. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 The last terms: 20
  • 24. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 The last terms: 20 Expanding the negative of the binomial product requires extra care.
  • 25. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 The last terms: 20 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.
  • 26. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 The last terms: 20 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – (3x – 4)(x + 5)
  • 27. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 The last terms: 20 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] Insert [ ]
  • 28. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 The last terms: 20 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20] Insert [ ] Expand
  • 29. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 The last terms: 20 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] Insert [ ] = – [ 3x2 + 15x – 4x – 20] Expand = – [ 3x2 + 11x – 20]
  • 30. Special Binomial Operations Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 The last terms: 20 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] Insert [ ] = – [ 3x2 + 15x – 4x – 20] Expand = – [ 3x2 + 11x – 20] = – 3x2 – 11x + 20 Remove [ ] and change signs.
  • 31. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL.
  • 32. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5)
  • 33. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) Distribute the sign.
  • 34. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 Distribute the sign. Expand
  • 35. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand
  • 36. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)
  • 37. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5)
  • 38. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20)
  • 39. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20
  • 40. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5
  • 41. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – (3x – 4)(x + 5)
  • 42. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets
  • 43. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 +11x – 20] Insert brackets Expand
  • 44. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets = 2x2 + x – 15 – [3x2 +11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Expand Remove brackets and combine
  • 45. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets = 2x2 + x – 15 – [3x2 +11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Expand Remove brackets and combine
  • 46. Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms.
  • 47. Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, (#x + #y)(#x + #y) = #x2 + #xy + #y2
  • 48. Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case.
  • 49. Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. Example D. Expand. (3x – 4y)(x + 5y)
  • 50. Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. Example D. Expand. (3x – 4y)(x + 5y) = 3x2 F
  • 51. Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. Example D. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx F OI
  • 52. Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. Example D. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx – 20y2 F OI L
  • 53. Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. Example D. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx – 20y2 = 3x2 + 11xy – 20y2
  • 54. Special Binomial Operations Exercise. A. Expand by FOIL method first. Then do them by inspection. 1. (x + 5)(x + 7) 2. (x – 5)(x + 7) 3. (x + 5)(x – 7) 4. (x – 5)(x – 7) 5. (3x – 5)(2x + 4) 6. (–x + 5)(3x + 8) 7. (2x – 5)(2x + 5) 8. (3x + 7)(3x – 7) 9. (–3x + 7)(4x + 3) 10. (–5x + 3)(3x – 4) 11. (2x – 5)(2x + 5) 12. (3x + 7)(3x – 7) 13. (9x + 4)(5x – 2) 14. (–5x + 3)(–3x + 1) 15. (5x – 1)(4x – 3) 16. (6x – 5)(–2x + 7) 17. (x + 5y)(x – 7y) 18. (x – 5y)(x – 7y) 19. (3x + 7y)(3x – 7y) 20. (–5x + 3y)(–3x + y) B. Expand and simplify. 21. –(2x – 5)(x + 3) 22. –(6x – 1)(3x – 4) 23. –(8x – 3)(2x + 1) 24. –(3x – 4)(4x – 3)
  • 55. Special Binomial Operations C. Expand and simplify. 25. (3x – 4)(x + 5) + (2x – 5)(x + 3) 26. (4x – 1)(2x – 5) + (x + 5)(x + 3) 27. (5x – 3)(x + 3) + (x + 5)(2x – 5) 28. (3x – 4)(x + 5) – (2x – 5)(x + 3) 29. (4x – 4)(2x – 5) – (x + 5)(x + 3) 30. (5x – 3)(x + 3) – (x + 5)(2x – 5) 31. (2x – 7)(2x – 5) – (3x – 1)(2x + 3) 32. (3x – 1)(x – 7) – (x – 7)(3x + 1) 33. (2x – 3)(4x + 3) – (x + 2)(6x – 5) 34. (2x – 5)2 – (3x – 1)2 35. (x – 7)2 – (2x + 3)2 36. (4x + 3)2 – (6x – 5)2