special binomial operations

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 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