Unit 6 - Objective 1 - Special Products

Some special products you will see:

common factor a (x + y) = ax + ay
difference of two squares (x + y)(x - y) = x2 - y2
perfect square (x + y)2 = (x + y)(x + y) = x2 + 2xy + y2
" (x - y)2 = (x - y)(x - y) = x2 - 2xy + y2
general product (x + a)(x + b) = x2 + (a + b) x + ab
perfect cube (x + y)3 = (x + y)(x + y)(x + y) = x3 + 3x2y + 3xy2 + y3
" (x - y)3 = (x - y)(x - y)(x - y) = x3 - 3x2y + 3xy2 - y3
sum of two cubes (x + y)(x2 - xy + y2) = x3 + y3
difference of two cubes (x - y)(x2 + xy + y2) = x3 - y3

 

Examples:

1. u6obj101.gif (599 bytes)  (use distributive property)
2. (x + 6) (x - 6)   (difference of two squares)
= (x)2 - (6)2
= x2 - 36
3. 3x (3x + 5y) (3x - 5y) 
= 3x ((3x)2 - (5y)2)
u6obj103.gif (1160 bytes)
    (Multiply the two binomials first because they are 
    the difference of two squares)

   (Now the distributive property)
4. (5a + 2b)2   (perfect square)
u6obj104.gif (2976 bytes)
= 25a2 + 10ab + 10ab + 4b2   (combine similar terms)
= 25a2 + 20ab + 4b2

or

using the perfect square formula this is the same as

(5a + 2b)2 = (5a)2 + 2 (5a) (2b) + (2b)2
=
25a2 + 20ab + 4b2
5. u6obj105.gif (4699 bytes)
6. (x + 2)3   (perfect cube)
u6obj106.gif (656 bytes)   (you could multiply the first 2 quantities)
= (x2 + 2x + 2x + 4) (x + 2)   (combine like terms)
= (x2 + 4x + 4) (x + 2)
= (x2) (x2) + (x2) (2) + (4x) (x) + (4x) (2) + (4) (x) + (4) (2)
= x3 + 2x2 + 4x2 + 8x + 4x + 8  (combine similar terms)
= x3 + 6x2 + 12x + 8

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