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Notes for Lesson 8-5: Factoring Special Products 8-5.1 – Recognizing and factoring perfect-square trinomials A perfect square trinomial is a trinomial where the first and last terms are perfect squares and the middle term is two times on factor from the first term and one factor from the last term. PERFECT-SQUARE TRINOMIAL a 2 2ab b 2 (a b)(a b) (a b) 2 EXAMPLE x 6 x 9 ( x 3)( x 3) ( x 3) 2 a 2 2ab b 2 (a b)(a b) (a b) 2 x 2 2 x 1 ( x 1)( x 1) ( x 1) 2 2 Examples: Determine whether each trinomial is a perfect square. If so, factor. x 2 12 x 36 4 x 2 12 x 9 x 2 9 x 16 ax a 2x b6 2ab 2( x)(6) 12 x b3 2ab 2(2 x)(3) 12 x ( x 6) 2 (2 x 3) 2 ax b4 2ab 1( x)(4) 8 x Not a perfect square 8-5.2 – Problem-Solving Applications The park in the center of the Place des Vosges in Paris, France, is in the shape of a square. The area of the park is 25 x 2 70 x 49 ft 2 . The side length of the park is in the form cx d , where c and d are whole numbers. Find the expression in terms of x for the perimeter of the park. Find the perimeter when x = 8 ft. Area Side 2 25 x 2 70 x 49 a 5x b7 2ab 2(5 x)(7) 70 x area (5 x 7) side 5 x 7 2 Perimeter 4( side) Perimeter 4(5 x 7) Perimeter 20 x 28 Perimeter if x 8 ft Perimeter 20(8) 28 Perimeter 160 28 Perimeter 188 ft 8-5.3 – Recognizing and factoring the difference of two squares. The difference of two squares occurs when you have a binomial where both terms are perfect squares and they are being subtracted from each other. DIFFERENCE OF TWO SQUARES a 2 b 2 (a b)(a b) EXAMPLE x 9 ( x 3)( x 3) 2 Examples: Determine whether each binomial is a difference of two squares. If so, factor. x 2 81 ax b9 ( x 9)( x 9) 9 p 4 16q 2 x6 7 y 2 a 3p2 b 4q a x3 (3 p 2 4q)(3 p 2 4q) Do Practice B #’s 2, 4, 5, 7, 8, 9 b 7 y 2 is not a perfect square Not the difference of two squares 1 4 x 2 Not subtraction Not the difference of two squares