Warm-Up: Factor the following polynomials 1.7x x – 5 1.x 2 – 15x x 4 – 8x x 6 1.6x 2 – 17x + 12

Identifying and factoring using special patterns Objective: factor polynomials using the difference of squares, sum/difference of cubes, and perfect square patterns

Review First we checked to see if there was a common factor among all terms Then we check and see if we can factor a quadratic using the simple method – we can only use simple factoring when “a” is 1 If “a” is NOT 1 we have to use the box method/AC method

Special Patterns Some polynomials can NOT be factored using any of these methods and have special patterns – Difference of Squares – Sum or difference of cubes – Perfect square

Difference of squares If there are 2 terms, look for special factoring patterns Difference of Squares: (a 2 – b 2 ) = (a – b)(a + b) – only works if you can take the square root of both terms 1.Set up the parenthesis ( ) ( ) 2.Put the first squared term in front of both 3.Put the second squared term at the end of both 4.Alternate the signs in between Example: (x 2 – 4)

Example: difference of squares 1.9x 2 – x 6 – y 8

Sums and differences of cubes If there are 2 terms, look for special factoring patterns Sum or Difference of Cubes: (MEMORIZE THESE!) – Only works if you can cube each term (a 3 + b 3 ) = (a + b)(a 2 – ab + b 2 ) (a 3 – b 3 ) = (a – b)(a 2 + ab + b 2 ) SOAP: SAME OPPOSITE ALWAYS POSITIVE 1.Label each term “a” and “b” 2.Find the cube of each term 3.Plug into the correct equation Example: x 3 – 8

Example: Sum and difference of cubes 1) 27x ) x 3 y 6 – 64

Perfect squares With 3 terms you might have a perfect square trinomial, which is the result of squaring a binomial: Saves some time to know a 2 + 2ab + b 2 = (a + b) 2 a 2 – 2ab + b 2 = (a – b) 2 1.Check to see if you can square root the first and last terms 2.If you can, find the squares and multiply them 3.Multiply the product by 2 (see if it gives you the middle term) If so then it is a perfect square Example: x x + 25

Example: perfect squares 3.16x x x 2 – 16x + 1

Practice: Special Patterns 1.x 2 – x 2 – x 2 – 25y x 4 – x 3 – – y 6 7.x 2 – 4x x 2 – 14x x 2 – 6x + 1

Standard Form Always put an expression in standard from before factoring – Set it equal to zero so that all terms are on one side of the equation

Example 1. 16x 2 = 8x

Example 2. x 2 + 7x = 18

Example 3. 2x 2 – 11x = -15

Practice Factor each of the following. 1.3x 2 – 20x – 7 = 0 1.2x 2 + 4x = 6 1.4x 2 – 17x = x x = -6 4.x 2 + 6x + 5 = x 2 – 5x = 3