Special Products and different techniques in factoring

1. SPECIAL
PRODUCTS AND
FACTORING

2. Special Products and
Factoring
Upon completion, you should be able to
• Find special products
• Factor a polynomial completely

3. Special Products
rules for finding products in a
faster way
long multiplication is the last
resort

6. Product of two binomials
 Examples:
1) (2x – 3)(x – 7)
    
2
ax b cx d acx ad bc x bd
     
2
2 14 3 21
x x x
   
2
2 17 21
x x
  

7. The product of two binomials with
like terms is equal to the product
of the first terms of the binomials
plus the sum of the products of the
two outer terms and two inner
terms plus the product of the last
terms of the binomials (FOIL
Method).

8. Multiply the following:
1. (2x+3) (x+8)
2. (4x+2) (3x-5)
3. (5x-2) (3x+7)

9. Square of a binomial
 Examples:
1) (2x – 3y)2
 
 
2 2 2
2 2 2
2
2
a b a ab b
a b a ab b
   
   
a b
      
2 2
2 2 2 3 3
x x y y
  
2 2
4 12 9
  
x xy y

10. The square of a binomial is
equal to the square of the first
term plus twice the product of
the two terms plus the square
of the second term.

11. Activity:
1. (X+6)2
2. (3x+9)2
3. (4y- 7)2
4. (2x-8y)2
5. (5x+6)2

12. Product of sum and
difference
 Examples:
1) (2x+3y2)(2x – 3y2)
   2 2
a b a b a b
   
   
x y
 
2
2 2
2 3
a b
x y
 
2 4
4 9

13. The product of the sum
and difference of two
terms is equal to the
square of the first term
minus the square of the
second term.

14. Activity:
1. (3x+y)(3x-y)
2. (4x+5)(4x-5)
3. (7y+8)(7y-8)
4. (3x+5)(3x-5)
5. (5x-8)(5x+8)

15. Cube of a binomial
 Examples:
1) (x + y2)3
2) (2x – 3y)3
 
3 3 2 2 3
3 3
a b a a b ab b
    
          
x x y x y y
   
2 3
3 2 2 2 2
3 3
x x y xy y
   
3 2 2 4 6
3 3
          
      
x x y x y y
2 3
3 2
2 3 2 3 3 2 3 3
   
x x y xy y
3 2 2 3
8 36 54 27

16.  The cube of a binomial is equal to the
cube of the first term plus thrice the
product of the second term and the
square of the first term plus thrice the
product of the first term and the
square of the second term plus the
cube of the second term.

17. Activity
1. (3x+4)3
2. (2x+3)3
3. (3x+5)3

18. Product of a binomial and a
trinomial
 Examples:
1) (x + y2)(x2 – xy2 + y4)
2) (2x – 3y)(4x2 + 6xy + 9y2)
3) (x – y)(x2 + 2xy + y2)
  
  
2 2 3 3
2 2 3 3
a b a ab b a b
a b a ab b a b
    
    
x y
 
3 6
   
x y
 
3
3
2 3 x y
 
3 3
8 27
# Pr
UseDistributive operty

19. The product of a binomial (x
+ y) and a trinomial (x2 – xy +
y2) is simply equal to the
sum of the cubes of the first
term and second term of the
binomial.

20. Factoring
 A polynomial with integer coefficients is
said to be prime if it has no monomial or
polynomial factors with integer
coefficients other than itself and one.
 A polynomial with integer coefficients is
said to be factored completely when each
of its polynomial factor is prime.

21.  The reverse of finding the related products
is a process called factoring. Rewriting a
polynomials as a product of polynomial
factors is call Factoring Polynomials. These
are:
1. Common Monomial Factor
2. Difference of two Squares
3. Factoring Trinomials
4. Factoring Perfect Square Trinomial
5. Factoring by Grouping

22. Factoring a common monomial factor
 Examples:
1) 4x2yz3– 2x4y3z2 +6 x3y2z4
 
ab ac a b c
  
x yz
 2 2
2  
z x y xyz
 
2 2 2
2 3

23. Difference of two squares
 Examples:
1) 4u2 – 9v2
2) (a+2b)2 – (3b + c)2
  
2 2
a b a b a b
   
  
u v u v
  
2 3 2 3
       
a b b c a b b c
         
  
2 3 2 3
  
a b c a b c
    
5

24. Factoring trinomials
 Examples:
1) x2 + 7x + 10
2) a2 – 10a + 24
3) a2 + 4ab – 21b2
4) 20x2 + 43xy + 14y2
  
2
( )
acx ad bc x bd ax b cx d
     
  
x x
  
5 2
  
a a
  
6 4
  
a b a b
  
7 3
  
x y x y
  
5 2 4 7
  
  
a a
8 3

25. Perfect square trinomials
 Examples:
1) y2 – 10y + 25
2) 24a3 + 72a2b + 54ab2
 
 
2
2 2
2
2 2
2
2
a ab b a b
a ab b a b
   
   
 
y
 
2
5
a
 6  
a ab b
 
2 2
4 12 9  
a a b
 
2
6 2 3

26. Sum and difference of cubes
 Examples:
1) 27 – x3
2) t3 + 8
  
  
3 3 2 2
3 3 2 2
a b a b a ab b
a b a b a ab b
    
    
 
x
 
3  
x x
  2
9 3
 
t
  2  
t x
 
2
2 4

27.  
y y
  
6 3
3 2 1
 
y
 
2
3
3 1
  
y y y
 
   
 
 
2
2
3 1 2 1
3) 3y6 – 6y3 + 3

28. Factoring by GROUPING
 Examples:
1) 10a3 + 25a – 4a2 – 10
   
a a a
    
3 2
10 25 4 10
 
 
a a2
5 2 5
 
 
a2
2 5
Common factor
 
 
a2
2 2 5
 

a
5 2

29. Factoring by GROUPING
 Examples:
2) 3xy – yz + 3xw – zw
 
 
xy xw
3 3
 
 
x y w
3
 
 
y w
 
  
yz zw
 
 
z y w
 

x z
3

30. Special Products and Factoring
Summary
 Always look for a common monomial
factor, FIRST.
 Factoring can also be done by grouping
some terms to yield a common
polynomial factor.

31.  Solve for the following special products.
 1. (3a + 2b) (3a + 2b) 6. (3x – 2y)3
 2. (a – 8) (a – 6) 7. (8x – 5 + 2y)2
 3. (2x + 5)3 8. (4c2 + 2d) (4c2 – d)
 4. (12k3 – 6) (12k3 + 6) 9. (g3 + h2)
 5. (7x + y) (7x – y) 10. (2x + 3y) (2x - 3y)

32. Factor the following
1. 4x2-9
2. 100-81x2
3. x2 – 7x + 12
4. 2x2 – xy – 55y2
5. 2xy + 6y + x +3
6. 9x2 + 6xy + y2
7. 8m3n3 – 125

33. Reflection
 When do we use special products?
 Enumerate the special products we
discussed in this unit.
 When is a polynomial completely
factored?
 Enumerate the different types of
factoring.