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

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)

a2 + b2 is Prime!!

Factoring Sums or Differences of Cubes

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

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

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

a) 27k3 - 125 = (3k)3  (5)3 27k3 - 125 = ( - )( + + ) 27k3 - 125 = (3k - 5)(9k2 + 15k + 25) b)

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

a) b) c) d) e)