2. Recall (a + b)(a – b) = a2 – b2 Perfect Square Variables x2 = (x)2
A variable raised to an even exponent

3. Example
Factor: a) x2  9 b) y2  16w2
c) 25a2  36b2
Solution
a) x2  9 = x2  32 = (x + 3) (x  3)
a2  b2 = (a + b) (a  b)
b) y2  16w2 = y2  (4w)2 = (y + 4w) (y  4w)
a2  b2 = (a + b) (a  b)
c) 25a2  36b2 = (5a)2 - (6b)2
= (5a + 6b)(5a – 6b)

4. a2 + b2 is Prime!!

5. Factoring Sums or Differences of Cubes

6. Perfect Cubes Numbers
Perfect Cubes Variables
x3 = (x)3
x6 = (x2)3
x9 = (x3)3
x12 = (x4)3
x15 = (x5)3
A variable raised to multiple of three

7. Write an equivalent expression by factoring: x3 + 27
Solution We observe that
x = x3 + (3)3
x = ( )( )
x = (x + 3)(x2 - 3x + 9)
Same
Opposite
Always Positive
= SOAP

8. Write an equivalent expression by factoring: x3  27
Solution We observe that
x = x3  (3)3
x = ( )( )
x = (x - 3)(x2 + 3x + 9)
Same
Opposite
Always Positive
= SOAP

9. a)
27k = (3k)3  (5)3
27k = ( )( )
27k = (3k - 5)(9k2 + 15k + 25) b)

10. (Note: The sum of two perfect squares
is prime!)

11. a) b) c) d) e)