The document discusses operations involving binomials and trinomials. It defines a binomial as a two-term polynomial of the form ax + b and a trinomial as a three-term polynomial of the form ax2 + bx + c. It states that the product of two binomials is a trinomial that can be found using the FOIL method: multiplying the first, outer, inner, and last terms. The FOIL method is demonstrated through examples multiplying binomial expressions. Expanding products involving negative binomials requires distributing the negative sign before using FOIL.

1. Special Binomial Operations

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 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,
Example D. Expand.
(3x – 4y)(x + 5y)
(#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 D. Expand.
(3x – 4y)(x + 5y)
= 3x2
(#x + #y)(#x + #y) = #x2 + #xy + #y2
The FOIL method is still applicable in this case.
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,
Example D. 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

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 D. 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

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 D. 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.

54. Multiplication Formulas

55. There are some important patterns in multiplying expressions
that it is worthwhile to memorize.
Multiplication Formulas

56. The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
There are some important patterns in multiplying expressions
that it is worthwhile to memorize.
Multiplication Formulas

57. The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
For example, the conjugate of (3x + 2) is (3x – 2),
There are some important patterns in multiplying expressions
that it is worthwhile to memorize.
Multiplication Formulas

58. The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
For example, the conjugate of (3x + 2) is (3x – 2), and
the conjugate of (2ab – c) is (2ab + c).
There are some important patterns in multiplying expressions
that it is worthwhile to memorize.
Multiplication Formulas

59. The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
For example, the conjugate of (3x + 2) is (3x – 2), and
the conjugate of (2ab – c) is (2ab + c).
Note: The conjugate is different from the opposite.
The opposite of (3x + 2) is (–3x – 2).
There are some important patterns in multiplying expressions
that it is worthwhile to memorize.
Multiplication Formulas

60. The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
There are some important patterns in multiplying expressions
that it is worthwhile to memorize.
I. Difference of Squares Formula
Multiplication Formulas
For example, the conjugate of (3x + 2) is (3x – 2), and
the conjugate of (2ab – c) is (2ab + c).
Note: The conjugate is different from the opposite.
The opposite of (3x + 2) is (–3x – 2).

61. The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
There are some important patterns in multiplying expressions
that it is worthwhile to memorize.
I. Difference of Squares Formula
(A + B)(A – B)
Conjugate Product
Multiplication Formulas
For example, the conjugate of (3x + 2) is (3x – 2), and
the conjugate of (2ab – c) is (2ab + c).
Note: The conjugate is different from the opposite.
The opposite of (3x + 2) is (–3x – 2).

62. The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
There are some important patterns in multiplying expressions
that it is worthwhile to memorize.
I. Difference of Squares Formula
(A + B)(A – B) = A2 – B2
Conjugate Product Difference of Squares
Multiplication Formulas
For example, the conjugate of (3x + 2) is (3x – 2), and
the conjugate of (2ab – c) is (2ab + c).
Note: The conjugate is different from the opposite.
The opposite of (3x + 2) is (–3x – 2).

63. The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
There are some important patterns in multiplying expressions
that it is worthwhile to memorize.
I. Difference of Squares Formula
(A + B)(A – B) = A2 – B2
To verify this :
(A + B)(A – B)
Conjugate Product Difference of Squares
Multiplication Formulas
For example, the conjugate of (3x + 2) is (3x – 2), and
the conjugate of (2ab – c) is (2ab + c).
Note: The conjugate is different from the opposite.
The opposite of (3x + 2) is (–3x – 2).

64. The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
There are some important patterns in multiplying expressions
that it is worthwhile to memorize.
I. Difference of Squares Formula
(A + B)(A – B) = A2 – B2
To verify this :
(A + B)(A – B) = A2 – AB + AB – B2
Conjugate Product Difference of Squares
Multiplication Formulas
For example, the conjugate of (3x + 2) is (3x – 2), and
the conjugate of (2ab – c) is (2ab + c).
Note: The conjugate is different from the opposite.
The opposite of (3x + 2) is (–3x – 2).

65. The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
There are some important patterns in multiplying expressions
that it is worthwhile to memorize.
I. Difference of Squares Formula
(A + B)(A – B) = A2 – B2
To verify this :
(A + B)(A – B) = A2 – AB + AB – B2
= A2 – B2
Conjugate Product Difference of Squares
Multiplication Formulas
For example, the conjugate of (3x + 2) is (3x – 2), and
the conjugate of (2ab – c) is (2ab + c).
Note: The conjugate is different from the opposite.
The opposite of (3x + 2) is (–3x – 2).

66. Multiplication Formulas
Here are some examples of squaring:

67. Multiplication Formulas
Here are some examples of squaring: (3x)2 =

68. Multiplication Formulas
Here are some examples of squaring: (3x)2 = 9x2,

69. Multiplication Formulas
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 =

70. Multiplication Formulas
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2,

71. Multiplication Formulas
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2

72. Multiplication Formulas
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

73. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2)
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

74. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2)
(A + B)(A – B)
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

75. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2
(A + B)(A – B) = A2 – B2
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

76. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

77. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

78. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

79. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

80. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas

81. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2

82. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2

83. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
We may check this easily by multiplying,

84. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
We may check this easily by multiplying,
(A + B)2 = (A + B)(A + B)

85. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
We may check this easily by multiplying,
(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2

86. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
We may check this easily by multiplying,
(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2

87. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
We may check this easily by multiplying,
(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2
We say that “(A + B)2 is A2, B2, plus twice A*B”,

88. Multiplication Formulas
Example E. Expand.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
We may check this easily by multiplying,
(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2
We say that “(A + B)2 is A2, B2, plus twice A*B”,
and “(A – B)2 is A2, B2, minus twice A*B”.

89. Example F.
a. (3x + 4)2
Multiplication Formulas

90. Example F.
a. (3x + 4)2
(A + B)2
Multiplication Formulas

91. Example F.
a. (3x + 4)2
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas

92. Example F.
a. (3x + 4)2 = (3x)2
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas

93. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4)
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas

94. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas

95. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas

96. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2

97. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

98. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2

99. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas

100. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.

101. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.
Example G. Calculate. Use the conjugate formula.
a. 51*49

102. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.
Example G. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1)

103. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.
Example G. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12

104. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.
Example G. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499

105. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.
Example G. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48

106. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.
Example G. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22

107. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.
Example G. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496

108. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.
Example G. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
c. 63*57 =

109. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.
Example G. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
c. 63*57 = (60 + 3)(60 – 3) = 602 – 32

110. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.
Example G. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591

111. Example F.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply.
The conjugate formula
(A + B)(A – B) = A2 – B2
may be used to multiply two numbers of the forms
(A + B) and (A – B) where A2 and B2 can be calculated easily.
Example G. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591

112. Multiplication Formulas
We observe the algebraic patterns:
(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...
…
(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
The Telescoping Products
These are telescoping products, the products compress into
two terms. In particular, we get the sum–of–powers formula:
(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
1 – r

113. Exercise. A. Calculate. Use the conjugate formula.
Multiplication Formulas
1. 21*19 2. 31*29 3. 41*39 4. 71*69
5. 22*18 6. 32*28 7. 52*48 8. 73*67
B. Calculate. Use the squaring formula.
9. 212 10. 312 11. 392 12. 692
13. 982 14. 30½2 15. 100½2 16. 49½2
18. (x + 5)(x – 5) 19. (x – 7)(x + 7)
20. (2x + 3)(2x – 3) 21. (3x – 5)(3x + 5)
C. Expand.
22. (7x + 2)(7x – 2) 23. (–7 + 3x )(–7 – 3x)
24. (–4x + 3)(–4x – 3) 25. (2x – 3y)(2x + 3y)
26. (4x – 5y)(5x + 5y) 27. (1 – 7y)(1 + 7y)
28. (5 – 3x)(5 + 3x) 29. (10 + 9x)(10 – 9x)
30. (x + 5)2 31. (x – 7)2
32. (2x + 3)2 33. (3x + 5y)2
34. (7x – 2y)2 35. (2x – h)2

114. 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)

115. 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

116. Ex. D. Multiply the following monomials.
1. 3x2(–3x2)
11. 4x(3x – 5) – 9(6x – 7)
Polynomial Operations
2. –3x2(8x5) 3. –5x2(–3x3)
4. –12( )
6
–5x3
5. 24( x3)
8
–5
6. 6x2( )
3
2x3
7. –15x4( x5)
5
–2
F. Expand and simplify.
E. Fill in the degrees of the products.
8. #x(#x2 + # x + #) = #x? + #x? + #x?
9. #x2(#x4 + # x3 + #x2) = #x? + #x? + #x?
10. #x4(#x3 + # x2 + #x + #) = #x? + #x? + #x? + #x?
12. –x(2x + 7) + 3(4x – 2)
13. –3x(3x + 2) – 8x(7x – 5) 14. 5x(–5x + 9) + 6x(6x – 1)
15. 2x(–4x + 2) – 3x(2x – 1) – 3(4x – 2)
16. –4x(–7x + 9) – 2x(2x – 5) + 9(4x + 2)

117. 18. (x + 5)(x + 7)
Polynomial Operations
G. Expand and simplify. (Use any method.)
19. (x – 5)(x + 7)
20. (x + 5)(x – 7) 21. (x – 5)(x – 7)
22. (3x – 5)(2x + 4) 23. (–x + 5)(3x + 8)
24. (2x – 5)(2x + 5) 25. (3x + 7)(3x – 7)
26. (3x2 – 5)(x – 6) 27. (8x – 2)(–4x2 – 7)
28. (2x – 7)(x2 – 3x + 9) 29. (5x + 3)(2x2 – x + 5)
38. (x – 1)(x + 1) 39. (x – 1)(x2 + x + 1)
40. (x – 1)(x3 + x2 + x + 1)
41. (x – 1)(x4 + x3 + x2 + x + 1)
42. What do you think the answer is for
(x – 1)(x50 + x49 + …+ x2 + x + 1)?
30. (x – 1)(x – 1) 31. (x + 1)2
32. (2x – 3)2 33. (5x + 4)2
34. 2x(2x – 1)(3x + 2) 35. 4x(3x – 2)(2x + 3)
36. (x – 5)(2x – 1)(3x + 2) 37. (2x + 1)(3x + 1)(x – 2)