Section 5.5 (Easy Factoring) Perfect Square Trinomials & Differences of Squares  Review: The Perfect Square Trinomial Rules (A + B) 2 = A 2 + 2AB + B 2 (A – B) 2 = A 2 – 2AB + B 2  If you see a trinomial that has these patterns, it factors easily: A 2 + 2AB + B 2 = (A + B) 2 A 2 – 2AB + B 2 = (A – B) 2  Some examples: x 2 – 2x + 1 = (x – 1) 2 r 2 + 6rs + 9s 2 = (r + 3s) 2  Some trinomials are more difficult to spot, so we need a reliable procedure … 5.51

Is the Trinomial a Perfect Square ?  Recall the square of two binomials pattern we used when multiplying: (b) 2 (±5) 2 (b – 5)

1: If necessary, Arrange in Descending Order  Why? Because we will need to check the 1 st and 3 rd terms, then check the middle term  x – 2x = x 2 – 2x + 1  25 – 10x + x 2 = x 2 – 10x + 25  6rs + 9s 2 + r 2 = r 2 + 6rs + 9s

2: Remove Any Common Factors (always check this before proceeding)  3x 2 – 15x + 12 = 3(x 2 – 5x + 4) = 3(x – 1)(x – 4)  Even when the 1 st and 3 rd terms are squares  16x x + 4 = 4(4x 2 + 4x + 1) = 4(2x + 1) 2  Sometimes a variable factors out  x 2 y – 10xy + 25y = y(x 2 – 10x + 25) = y(x – 5)

3: See if the 1 st and 3 rd Terms are Squares  Check 36x x + 49  (6x) 2 … (7) 2 ok, might be!  Check 9x 2 – 68xy + 121y 2  (3x) 2 … (11y) 2 ok, might be!  Check 16a a + 63  (4a) 2 … (7)(9) no, can’t be!  Ok, let’s see if the middle terms are right 5.55

 Check 36x x + 49  (6x) 2 … (7) 2 ok, might be!  2(6x)(7) = 84x yes, it is (6x + 7) 2  Check 9x 2 – 68xy + 121y 2  (3x) 2 … (-11y) 2 ok, might be!  2(3x)(-11y) = -66xy ≠ -68xy no, not a PST  Check 16a 2 – 72a + 81  (4a) 2 … (-9) 2 ok, might be!  2(4a)(-9) = -72a yes, it is (4a – 9) 2 4: See if the Middle Term is 2AB 5.56

 x 2 + 8x + 16 = (x + 4) 2  (x) 2 (4) 2 2(x)(4) = 8x yes, it matches   t 2 – 5t + 4 = not a PST… but it factors: (t - 1)(t - 4)  (t) 2 (-2) 2 2(t)(-2) = -4t no, it’s not -5t   25 + y y = (y + 5) 2  y y + 25 descending order  (y) 2 (5) 2 2(y)(5) = 10y yes, it matches   3x 2 – 15x + 27 = not a PST  3(x 2 – 5x + 9) remove common factor  (x) 2 (-3) 2 2(x)(-3) = -6x no, it’s not -5x Are These Perfect Square Trinomials? PST Tests: 1. Descending Order 2. Common Factors 3. 1 st and 3 rd Terms (A) 2 and (B) 2 4. Middle Term 2AB or -2AB 5.57

Difference of Squares Binomials  Remember that the middle term disappears? (A + B)(A – B) = A 2 - B 2  It’s easy factoring when you find binomials of this pattern A 2 – B 2 = (A + B)(A – B)  Examples: x 2 – 9 = (x) 2 – (3) 2 = (x + 3)(x – 3) 4t 2 – 49 = (2t) 2 – (7) 2 = (2t + 7)(2t – 7) a 2 – 25b 2 = two variables squared (a) 2 – (5b) 2 = (a + 5b)(a – 5b) 18 – 2y 4 = constant 1 st, variable square 2 nd 2 [ (3) 2 – (y 2 ) 2 ] = 2(3 + y 2 )(3 – y 2 ) 5.58

More Difference of Squares  Examples: x 2 – 1/9 = perfect square fractions (x) 2 – (⅓) 2 = (x + ⅓)(x – ⅓) 18x 2 – 50x 4 = common factors must be removed 2x 2 [ 9 – 25x 2 ] = 2x 2 [ (3) 2 – (5x) 2 ] = 2x 2 (3 + 5x)(3 – 5x) p 8 – 1 = factor completely (p 4 ) 2 – (1) 2 = (p 4 + 1)(p 4 – 1) another difference of 2 squares (p 4 + 1)(p 2 + 1)(p 2 – 1) and another (p 4 + 1)(p 2 + 1)(p + 1)(p – 1) 5.59

What Next?  Section 5.6 – Factoring Sums & Differences of Cubes