7 special binomial operations and formulas

Special Binomial Operations

A binomial is a two-term polynomial. Special Binomial Operations

A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.


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.


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.


A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c.The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + #




F: To get the x2-term, multiply the two Front x-terms of the binomials.



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.



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.



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.



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.

Example A. Multiply using FOIL method.a. (x + 3)(x – 4)


Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2


The front terms: x2-term



Outer pair: –4x



Inner pair: –4x + 3x

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x


Outer Inner pairs: –4x + 3x = –x

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12


The last terms: –12


b. (3x + 4)(–2x + 5)




b. (3x + 4)(–2x + 5) = –6x2

The front terms: –6x2




b. (3x + 4)(–2x + 5) = –6x2

Outer pair: 15x




b. (3x + 4)(–2x + 5) = –6x2

Inner pair: 15x – 8x




b. (3x + 4)(–2x + 5) = –6x2 + 7x

Outer and Inner pair: 15x – 8x = 7x




b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20


The last terms: 20



b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20


The last terms: 20


Expanding the negative of the binomial product requires extra care.


b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20


The last terms: 20


Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.


b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20


The last terms: 20


Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.Example B. Expand.a. – (3x – 4)(x + 5)


b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20


The last terms: 20


Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.Example B. Expand.a. – [(3x – 4)(x + 5)] Insert [ ]


b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20


The last terms: 20


Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.Example B. Expand.a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20]

Insert [ ]Expand


b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20


The last terms: 20


Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.Example B. Expand.a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20] = – [ 3x2 + 11x – 20]

Insert [ ]Expand


b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20


The last terms: 20


Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.Example B. Expand.a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20] = – [ 3x2 + 11x – 20] = – 3x2 – 11x + 20

Insert [ ]Expand

Remove [ ] andchange signs.

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.

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)

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.

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

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




b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)



b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)= (2x – 5)(x +3) + (–3x + 4)(x + 5)



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)



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



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



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)



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



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



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



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

Special Binomial OperationsIf the binomials are in x and y, then the products consist of the x2, xy and y2 terms.

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


(#x + #y)(#x + #y) = #x2 + #xy + #y2

The FOIL method is still applicable in this case.


Example D. Expand.(3x – 4y)(x + 5y)

(#x + #y)(#x + #y) = #x2 + #xy + #y2



Example D. Expand.(3x – 4y)(x + 5y)= 3x2

(#x + #y)(#x + #y) = #x2 + #xy + #y2


F


Example D. Expand.(3x – 4y)(x + 5y)= 3x2 + 15xy – 4yx

(#x + #y)(#x + #y) = #x2 + #xy + #y2


F OI


Example D. Expand.(3x – 4y)(x + 5y)= 3x2 + 15xy – 4yx – 20y2

(#x + #y)(#x + #y) = #x2 + #xy + #y2


F OI L


Example D. Expand.(3x – 4y)(x + 5y)= 3x2 + 15xy – 4yx – 20y2

= 3x2 + 11xy – 20y2

(#x + #y)(#x + #y) = #x2 + #xy + #y2


Multiplication Formulas

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


The two binomials (A + B) and (A – B) are said to be the conjugate of each other.



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),



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).



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).





I. Difference of Squares Formula


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).



I. Difference of Squares Formula (A + B)(A – B)

Conjugate Product





I. Difference of Squares Formula (A + B)(A – B) = A2 – B2

Conjugate Product Difference of Squares






To verify this :(A + B)(A – B)







To verify this :(A + B)(A – B) = A2 – AB + AB – B2







To verify this :(A + B)(A – B) = A2 – AB + AB – B2

= A2 – B2




Multiplication FormulasHere are some examples of squaring:

Multiplication FormulasHere are some examples of squaring: (3x)2 =

Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2,

Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2, (2xy)2 =

Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2,

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

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


Example D. Expand.a. (3x + 2)(3x – 2)

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


Example D. Expand.a. (3x + 2)(3x – 2)

(A + B)(A – B)



Example D. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2

(A + B)(A – B) = A2 – B2



Example D. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2




(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2)




(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2





= 4x2y2 – 25z4





= 4x2y2 – 25z4


II. Square Formulas




= 4x2y2 – 25z4


II. Square Formulas (A + B)2 = A2 + 2AB + B2




= 4x2y2 – 25z4



(A – B)2 = A2 – 2AB + B2




= 4x2y2 – 25z4



(A – B)2 = A2 – 2AB + B2

We may check this easily by multiplying,




= 4x2y2 – 25z4



(A – B)2 = A2 – 2AB + B2

We may check this easily by multiplying,(A + B)2 = (A + B)(A + B)




= 4x2y2 – 25z4



(A – B)2 = A2 – 2AB + B2

We may check this easily by multiplying,(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2




= 4x2y2 – 25z4



(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




= 4x2y2 – 25z4



(A – B)2 = A2 – 2AB + B2


We say that “(A + B)2 is A2, B2, plus twice A*B”,




= 4x2y2 – 25z4



(A – B)2 = 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”.

Example E.a. (3x + 4)2



(A + B)2



(A + B)2 = A2 + 2AB + B2


Example E.a. (3x + 4)2 = (3x)2

(A + B)2 = A2 + 2AB + B2


Example E.a. (3x + 4)2 = (3x)2 + 2(3x)(4)

(A + B)2 = A2 + 2AB + B2


Example E.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42

(A + B)2 = A2 + 2AB + B2


Example E.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2


Example B.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2


b. (3a – 5b)2

Example E.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2


b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

Example E.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2


b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

B. Expand and simplify.


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)

C. Expand and simplify.25. (3x – 4)(x + 5) + (2x – 5)(x + 3)26. (4x – 1)(2x – 5) + (x + 5)(x + 3)27. (5x – 3)(x + 3) + (x + 5)(2x – 5)


28. (3x – 4)(x + 5) – (2x – 5)(x + 3)29. (4x – 4)(2x – 5) – (x + 5)(x + 3)30. (5x – 3)(x + 3) – (x + 5)(2x – 5)31. (2x – 7)(2x – 5) – (3x – 1)(2x + 3)32. (3x – 1)(x – 7) – (x – 7)(3x + 1)33. (2x – 3)(4x + 3) – (x + 2)(6x – 5)34. (2x – 5)2 – (3x – 1)2

35. (x – 7)2 – (2x + 3)2

36. (4x + 3)2 – (6x – 5)2


18. (x + 5)(x – 5) 19. (x – 7)(x + 7)20. (2x + 3)(2x – 3) 21. (3x – 5)(3x + 5)

D. 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

Exercise. E. Calculate. Use the conjugate formula.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

Exercise. E. 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

Education

7 special binomial operations and formulas