MODULE 2: SPECIAL PRODUCTS AND BINOMIAL THEOREM
Objectives:
At the end of this module, the students will be able to:
1. Find the product of the following algebraic expressions.
2. Use special products to expand the following algebraic expressions
3. Expand by using binomial theorem.
Introduction:
In the previous module, we learned and applied rules and procedures in finding the
products of algebraic expressions. In this module, we will familiarize ourselves with
certain products of algebraic expressions that occur frequently and show patterns in
their solutions, which are commonly called special products.
BY recognizing these special products and applying the corresponding special
formulas, we will be able to shorten our procedures and contribute to the speed and
accuracy of algebraic computations.
Pre-test:
For number,s 1-3, find the product of the following:
1. 2 x 2 32 x 2 3
2. x y z x y z
3. m 2 8n
3
For numbers 3-5, expand the following using the binomial theorem:
4. x 35
5. 1 2ab
8
Pre-test Feedback:
1. 4 x 4 9
2. x 2 y 2 2 xy z 2
3. m 6 24m 4 n 192m 2 n 2 512n 3
4. x 5 15 x 4 90 x 3 270 x 2 405 x 243
5. 1 16ab 112a 2b 2 448a 3b 3 1120 a 4b 4 1792 a 5b 5 1792 a 6b 6 1024 a 7 b 7 256a 8b 8
Special product - product of algebraic expressions which can be obtained in one or two
steps without doing long multiplication. It facilitates the simplification of algebraic
expressions.
Special products that are frequently used:
1. Product of a sum and difference of two terms
Form: (a b)(a b) a 2 b 2
Example: (8 x 2 y )(8 x 2 y ) 64 x 2 4 y 2
2. Square of a binomial
Form: (a b) 2 a 2 2ab b 2
Example: (2 x 3 y ) 2 4 x 2 12 xy 9 y 2
3. Cube of a binomial
Form: (a b) 3 a 3 3a 2b 3ab 2 b 3
Example: ( x 3 y ) 3 x 3 9 x 2 y 27 xy 2 27 y 3
4. Product of two dissimilar binomials
Form: (a b)(c d ) ac ad bc bd
Example: ( w x)( y z ) wy wz xy xz
5. Square of a polynomial
Form: (a b c) 2 a 2 b 2 c 2 2ab 2bc 2ac
Example: (2 x 3 y 4 z ) 2 4 x 2 9 y 2 16 z 2 12 xy 24 yz 16 xz
6. Expansion of (a b) n
Properties:
a. There is always (n+1) terms in the product.
b. The first term is always a n and the last term b n .
c. If the sign in the binomial is positive, all the terms in the expansion will be positive,
however if the sign is negative, the terms in the expansion will consist of alternate + and
- signs.
d. The power of a are in descending order while the powers of b are in ascending
order
e. To get the coefficient of the terms in the simplified form, start always with the second
term . Using the preceding term as a basis, multiply the exponent of a by its numerical
coefficient and then divide the product by the exponent of b in the term being
considered.
Example: (a b) 5 a 5 5a 4b 10a 3b 2 10a 2b 3 5ab 4 b 5
Practice Exercise A: Use special products to find the product of the following.
1. ( x 5)( x 3)
2. (3 x 1)(2 x 5)
3. (3 x 5) 2
4. (5 x 2 3u 2 ) 2
5. (2 x 3)(2 x 3)
6. (3 x 4 y )(3 x 4 y )
7. (5 x 3 y ) 3
8. (2 x 2 5 y 3 )(2 x 2 5 y 3 )
9. x y z
2
10. x 2 2 x 3 x 2 2 x 3
11. 2 xy x y
2 2 2 7
12. 3x 4 y 3 6
13. ax 2bx 4a
2 2
14. 3x 2 y 4 z 2 x y 2 z
2 3 2 3
15. x 3 y 2 2 x 3 y 2 2
16. 3ab 2 4a 3b
2
17. 3a 3b 2c 5
3
18. 3 x 2 2 y 3 4 z 2
2
19. 2a 3b
4
20. 5a 3b 4ab 2
3
Practice Exercise B: Without performing long multiplication, obtain the following:
1. (2b 5)(2b 7)
2. x 2 y 3 z
2
3. 6 xy 4 z 6 xy 4 z
4. y 53 y 7
5. (4 2 z 3 z ) 2
6. 3 x 6
2
7. a b c a b c
8. 2 x 10
3
9. 3 x 2 2
3
10. m 3n m 2 3mn 9n 2
11. y 2 3x 2
6
12. x 2 y 2
6
13. bw 2 cx 3 dy
2
14. a b
4
15. 2 x 2 y 2 52 x 2 y 2 5
16. 3c 2 4d 3
2
17. a 2b c 4
3
18. x 2 y 3 z 4
2
19. 2 5ab
4
20. xy 2z
3
Post test:
For number,s 1-3, find the product of the following:
1. 2 x 2 32 x 2 3
2. x y z x y z
3. m 2 8n
3
For numbers 3-5, expand the following using the binomial theorem:
1. x 35
2. 1 2ab
8
Post test Feedback:
1. 4 x 4 9
2. x 2 y 2 2 xy z 2
3. m 6 24m 4 n 192m 2 n 2 512n 3
4. x 5 15 x 4 90 x 3 270 x 2 405 x 243
5. 1 16ab 112a 2b 2 448a 3b 3 1120 a 4b 4 1792 a 5b 5 1792 a 6b 6 1024 a 7 b 7 256a 8b 8
