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- 1. Special Binomial Operations Back to 123a-Home
- 2. A binomial is a two-term polynomial. Special Binomial Operations
- 3. A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. Special Binomial Operations
- 4. 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. Special Binomial Operations
- 5. 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. Special Binomial Operations
- 6. 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 + # Special Binomial Operations
- 7. 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 + # Special Binomial Operations F: To get the x2-term, multiply the two Front x-terms of the binomials.
- 8. 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 + # Special Binomial Operations 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. 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 + # Special Binomial Operations 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. 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 + # Special Binomial Operations 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. 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 + # Special Binomial Operations 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. Example A. Multiply using FOIL method. a. (x + 3)(x – 4) Special Binomial Operations
- 13. Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 Special Binomial Operations The front terms: x2-term
- 14. Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 Special Binomial Operations Outer pair: –4x
- 15. Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 Special Binomial Operations Inner pair: –4x + 3x
- 16. Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x Special Binomial Operations Outer Inner pairs: –4x + 3x = –x
- 17. Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 Special Binomial Operations The last terms: –12
- 18. Special Binomial Operations b. (3x + 4)(–2x + 5) Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
- 19. Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 The front terms: –6x2 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
- 20. Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 Outer pair: 15x Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
- 21. Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 Inner pair: 15x – 8x Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
- 22. Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x Outer and Inner pair: 15x – 8x = 7x Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
- 23. Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12
- 24. Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care.
- 25. Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 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 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 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 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 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 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 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 b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 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] = – [ 3x2 + 11x – 20] Insert [ ] Expand
- 30. Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 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] = – [ 3x2 + 11x – 20] = – 3x2 – 11x + 20 Insert [ ] Expand Remove [ ] and change all the signs.
- 31. Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 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] = – [ 3x2 + 11x – 20] = – 3x2 – 11x + 20 Insert [ ] Expand Remove [ ] and change all the signs. The key here is that all three terms change signs.
- 32. Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL.
- 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)
- 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) Distribute the sign.
- 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 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
- 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 Below we present both versions of the algebra for simplifying the differences of two products of binomials.
- 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 Example D. Expand and simplify. Below we present both versions of the algebra for simplifying the differences of two products of binomials. a. (2x – 5)(x +3) – [(3x – 4)(x + 5)]
- 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 Example D. Expand and simplify. Below we present both versions of the algebra for simplifying the differences of two products of binomials. a. (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets
- 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 Example D. Expand and simplify. Below we present both versions of the algebra for simplifying the differences of two products of binomials. a. (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets = 2x2 + x – 15 – [3x2 +11x – 20] Expand
- 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 Example D. Expand and simplify. Below we present both versions of the algebra for simplifying the differences of two products of binomials. a. (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets = 2x2 + x – 15 – [3x2 +11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 Expand Remove brackets and combine
- 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 Example D. Expand and simplify. Below we present both versions of the algebra for simplifying the differences of two products of binomials. a. (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
- 43. Special Binomial Operations b. Expand and simplify. (2x – 5)(x +3) – (3x – 4)(x + 5)
- 44. Special Binomial Operations b. Expand and simplify. (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) Distribute the “–” sign
- 45. Special Binomial Operations b. Expand and simplify. (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 – 3x2 –15x + 4x + 20 Distribute the “–” sign Expand
- 46. Special Binomial Operations b. Expand and simplify. (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 Distribute the “–” sign Expand
- 47. Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms.
- 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
- 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.
- 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, Example E. Expand. (3x – 4y)(x + 5y) (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case.
- 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, Example E. Expand. (3x – 4y)(x + 5y) = 3x2 (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. F OI L
- 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, Example E. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. F OI
- 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, Example E. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx – 20y2 (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. F OI L
- 54. Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, Example E. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx – 20y2 = 3x2 + 11xy – 20y2 (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. F OI L
- 55. B. Expand and simplify. Special Binomial Operations 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) Exercise. A. Expand by FOIL method first. Then do them by inspection. 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) 21. –(2x – 5)(x + 3) 22. –(6x – 1)(3x – 4) 23. –(8x – 3)(2x + 1) 24. –(3x – 4)(4x – 3)
- 56. 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) Special Binomial Operations 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

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