Math A-100 IUSB Chapter 6.4 - 6.5 detailed notes – Week 10 Factoring 2 terms - Special Cases in Factoring If you have only 2 terms, it is usually a Special Case. Special Case #1 – Difference of Squares (sound familiar from Chapter 5???) If you recognize that you have the Difference of Squares, you can go straight to the answer. REMEMBER: (x + 8)(x – 8) = (x)(x) – 8x + 8x – (8)(8)  x2 – 64 So, if you are given x2 – 64  you can go right back to the ( )’s  (x + 8)(x – 8) For Difference of Squares – basic answer is (sqrt of a + sqrt of c)( sqrt of a - sqrt of c) in either order Example 1: x2 – 81 X2 is a square, its sqrt = x 81 is a square, its sqrt = 9 Answer (x – 9)(x + 9) Example 2: 25x2 – 16 25x2 is a square, its sqrt = 5x 16 is a square, its sqrt = 4 Answer (5x – 4)(5x + 4) Example 3: x4 – 625 x4 is a square, its sqrt = x2 625 is a square, its sqrt = 25 Answer (x2 – 25)(x2 +25) BUT WAIT – you are not done yet!! Isn’t x2 – 25 also the difference of squares? x2 – 25 = (x – 5)(x + 5) Final answer: (x – 5)(x + 5)(x2 +25) NOTE: the “”ac” method also works here. Just remember that b = 0 Redoing Example 1: x2 – 81 Steps: 1) a = 1, b = 0, & c = - 81 SHORT CUT is available here 2) a*c = 1*(- 81) = - 81 3) What 2 numbers multiply to you – 81 and add up to 0? (-9)(9) = -81 & -9 + 9 = 0; our numbers are -9 + 9 Short Cut - go right to the answer: (x – 9)(x + 9) Special Case #2 – The Sum of Squares We just saw that the Difference of Squares factors (i.e. breaks down). What about the Sum of Square? For example: x2 + 16 (x + 4)(x - 4) = x2 - 16  this does not work! (x + 4)(x + 4) = x2 + 8x + 16  this does not work either! (x - 4)(x - 4) = x2 - 8x + 16  this does not work! BOTTOM LINE: x2 + 16 does not factor  PRIME Sec 6.5 – is a collection of the various factoring questions from Sec 6.1 – 6.4