4 6multiplication formulas

Multiplication Formulas
Frank Ma © 2011

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.

For example, the conjugate of (3x + 2) is (3x – 2),

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).

I. Difference of Squares Formula

(A + B)(A – B)
Conjugate Product

(A + B)(A – B) = A2 – B2
Conjugate Product
Difference of Squares

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

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

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

Here are some examples of squaring:

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

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

(2xy)2 =

(2xy)2 = 4x2y2,

(2xy)2 = 4x2y2, and (5z2)2

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(3x + 2)(3x – 2)

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(3x + 2)(3x – 2)
(A + B)(A – B)

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(3x + 2)(3x – 2) = (3x)2 – (2)2
(A + B)(A – B) = A2 – B2

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(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

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(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

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(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
II. Square Formulas

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(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
II. Square Formulas
(A + B)2 = A2 + 2AB + B2

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(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
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(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
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
We may check this easily by multiplying,

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(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
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
(A + B)2 = (A + B)(A + B)

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(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
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(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
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(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
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
We say that “(A + B)2 is A2, B2, plus twice A*B”,

(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Example A. Expand.
(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
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(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 B.
(3x + 4)2

Example B.
(3x + 4)2
(A + B)2

Example B.
(3x + 4)2
(A + B)2 = A2 + 2AB + B2

Example B.
(3x + 4)2 = (3x)2
(A + B)2 = A2 + 2AB + B2

Example B.
(3x + 4)2 = (3x)2 + 2(3x)(4)
(A + B)2 = A2 + 2AB + B2

Example B.
(3x + 4)2 = (3x)2 + 2(3x)(4) + 42
(A + B)2 = A2 + 2AB + B2

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

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

Example B.
(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 B.
(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

Example B.
(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
III. Some Applications of the Formulas

Example B.
(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
We can use the above formulas to help us multiply.

Example B.
(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
Example C. Calculate. Use the conjugate formula.
a. 51*49

Example B.
(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
a. 51*49 = (50 + 1)(50 – 1)

Example B.
(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
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12

Example B.
(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
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499

Example B.
(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
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48

Example B.
(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
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22

Example B.
(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
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

Example B.
(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
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 =

Example B.
(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
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

Example B.
(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
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

Example B.
(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
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
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.

The Squaring Formulas.
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.

Example D. Calculate. Use the squaring formulas.
a. 512

a. 512 = (50 + 1)2

a. 512 = (50 + 1)2 = 502 + 12

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601
b. 492

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601
b. 492 = (50 – 1)2

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601
b. 492 = (50 – 1)2 = 502 + 12

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601
b. 492 = (50 – 1)2 = 502 + 12– 2(50)(1)

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601
b. 492 = (50 – 1)2 = 502 + 12– 2(50)(1)
= 2,500 + 1 – 100

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601
b. 492 = (50 – 1)2 = 502 + 12– 2(50)(1)
= 2,500 + 1 – 100
= 2,401

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601
b. 492 = (50 – 1)2 = 502 + 12– 2(50)(1)
= 2,500 + 1 – 100
= 2,401
b. (50½) 2 = (50 + ½ )2

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601
b. 492 = (50 – 1)2 = 502 + 12– 2(50)(1)
= 2,500 + 1 – 100
= 2,401
b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601
b. 492 = (50 – 1)2 = 502 + 12– 2(50)(1)
= 2,500 + 1 – 100
= 2,401
b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2+ 2 (½) (50)

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601
b. 492 = (50 – 1)2 = 502 + 12– 2(50)(1)
= 2,500 + 1 – 100
= 2,401
b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2+ 2 (½) (50)
= 2,500 + 1/4 + 50

a. 512 = (50 + 1)2 = 502 + 12+ 2(50)(1)
= 2,500 + 1 + 100
= 2,601
b. 492 = (50 – 1)2 = 502 + 12– 2(50)(1)
= 2,500 + 1 – 100
= 2,401
b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2+ 2 (½) (50)
= 2,500 + 1/4 + 50
= 2,550¼

4 6multiplication formulas

by math123a on Jun 22, 2011

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4 6multiplication formulas — Presentation Transcript