47 operations of 2nd degree expressions and formulas
TRANSCRIPT
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Special Binomial Operations
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A binomial is a two-term polynomial. Special Binomial Operations
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A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.
Special Binomial Operations
![Page 4: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/4.jpg)
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
![Page 5: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/5.jpg)
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
![Page 6: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/6.jpg)
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
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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.
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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.
![Page 9: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/9.jpg)
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.
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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.
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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.
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Example A. Multiply using FOIL method.a. (x + 3)(x – 4)
Special Binomial Operations
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Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2
Special Binomial Operations
The front terms: x2-term
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Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2
Special Binomial Operations
Outer pair: –4x
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Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2
Special Binomial Operations
Inner pair: –4x + 3x
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Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x
Special Binomial Operations
Outer Inner pairs: –4x + 3x = –x
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Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12
Special Binomial Operations
The last terms: –12
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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
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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
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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
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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
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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
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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
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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.
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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.
![Page 26: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/26.jpg)
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)
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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 [ ]
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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
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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
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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 [ ] andchange signs.
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Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.
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Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5)
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Special Binomial OperationsAnother 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.
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Special Binomial OperationsAnother 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
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Special Binomial OperationsAnother 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
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Special Binomial OperationsAnother 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)
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Special Binomial OperationsAnother 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)
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Special Binomial OperationsAnother 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)
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Special Binomial OperationsAnother 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
![Page 40: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/40.jpg)
Special Binomial OperationsAnother 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
![Page 41: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/41.jpg)
Special Binomial OperationsAnother 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)
![Page 42: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/42.jpg)
Special Binomial OperationsAnother 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
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Special Binomial OperationsAnother 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 bracketsExpand
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Special Binomial OperationsAnother 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
ExpandRemove brackets and combine
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Special Binomial OperationsAnother 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
ExpandRemove brackets and combine
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Special Binomial OperationsIf the binomials are in x and y, then the products consist of the x2, xy and y2 terms.
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Special Binomial OperationsIf 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
![Page 48: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/48.jpg)
Special Binomial OperationsIf 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.
![Page 49: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/49.jpg)
Special Binomial OperationsIf 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.
![Page 50: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/50.jpg)
Special Binomial OperationsIf 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
![Page 51: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/51.jpg)
Special Binomial OperationsIf 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
![Page 52: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/52.jpg)
Special Binomial OperationsIf 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
![Page 53: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/53.jpg)
Special Binomial OperationsIf 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.
![Page 54: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/54.jpg)
Multiplication Formulas
![Page 55: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/55.jpg)
There are some important patterns in multiplying expressions that it is worthwhile to memorize.
Multiplication Formulas
![Page 56: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/56.jpg)
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
![Page 57: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/57.jpg)
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
![Page 58: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/58.jpg)
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), andthe conjugate of (2ab – c) is (2ab + c).
There are some important patterns in multiplying expressions that it is worthwhile to memorize.
Multiplication Formulas
![Page 59: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/59.jpg)
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), andthe 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
![Page 60: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/60.jpg)
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), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
![Page 61: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/61.jpg)
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), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
![Page 62: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/62.jpg)
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), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
![Page 63: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/63.jpg)
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), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
![Page 64: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/64.jpg)
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), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
![Page 65: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/65.jpg)
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), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
![Page 66: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/66.jpg)
Multiplication FormulasHere are some examples of squaring:
![Page 67: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/67.jpg)
Multiplication FormulasHere are some examples of squaring: (3x)2 =
![Page 68: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/68.jpg)
Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2,
![Page 69: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/69.jpg)
Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2, (2xy)2 =
![Page 70: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/70.jpg)
Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2,
![Page 71: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/71.jpg)
Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2
![Page 72: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/72.jpg)
Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
![Page 73: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/73.jpg)
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.
![Page 74: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/74.jpg)
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.
![Page 75: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/75.jpg)
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.
![Page 76: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/76.jpg)
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.
![Page 77: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/77.jpg)
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.
![Page 78: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/78.jpg)
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.
![Page 79: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/79.jpg)
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.
![Page 80: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/80.jpg)
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
![Page 81: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/81.jpg)
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
![Page 82: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/82.jpg)
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
![Page 83: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/83.jpg)
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,
![Page 84: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/84.jpg)
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)
![Page 85: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/85.jpg)
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
![Page 86: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/86.jpg)
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
![Page 87: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/87.jpg)
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”,
![Page 88: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/88.jpg)
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”.
![Page 89: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/89.jpg)
Example F.a. (3x + 4)2
Multiplication Formulas
![Page 90: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/90.jpg)
Example F.a. (3x + 4)2
(A + B)2
Multiplication Formulas
![Page 91: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/91.jpg)
Example F.a. (3x + 4)2
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
![Page 92: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/92.jpg)
Example F.a. (3x + 4)2 = (3x)2
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
![Page 93: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/93.jpg)
Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4)
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
![Page 94: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/94.jpg)
Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
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Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
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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
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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
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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
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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
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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 FormulasWe can use the above formulas to help us multiply.
![Page 101: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/101.jpg)
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 FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49
![Page 102: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/102.jpg)
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 FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49 = (50 + 1)(50 – 1)
![Page 103: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/103.jpg)
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 FormulasWe 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
![Page 104: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/104.jpg)
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 FormulasWe 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
![Page 105: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/105.jpg)
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 FormulasWe 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 = 2,499b. 52*48
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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 FormulasWe 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 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22
![Page 107: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/107.jpg)
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 FormulasWe 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 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
![Page 108: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/108.jpg)
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 FormulasWe 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 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496c. 63*57 =
![Page 109: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/109.jpg)
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 FormulasWe 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 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496c. 63*57 = (60 + 3)(60 – 3) = 602 – 32
![Page 110: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/110.jpg)
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 FormulasWe 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 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591
![Page 111: 47 operations of 2nd degree expressions and formulas](https://reader036.vdocuments.us/reader036/viewer/2022062523/58e94f4e1a28ab262c8b59b3/html5/thumbnails/111.jpg)
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 FormulasWe 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 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591
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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
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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
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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)
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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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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)
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18. (x + 5)(x + 7)
Polynomial OperationsG. 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)