Example: Factor the polynomial 21x2 – 41x + 10. No GCF Puzzle pieces for 21x2 x, 21x 3x, 7x Puzzle pieces for 10 1, 10 2, 5 We know the signs ( – )( – ) First guess (3x – 2)(7x – 5) Middle term – 14x – 15x = –29x Second guess (3x – 5)(7x – 2) Middle term – 35x – 6x = –41x 21x2 – 41x + 10 = (3x – 5)(7x – 2)

Example: Factor the polynomial 25x2 + 20x + 4. No GCF Puzzle pieces for 25x2 x, 25x 5x, 5x Puzzle pieces for 4 1, 4 2, 2 We know the signs ( + )( + ) First guess (5x + 2)(5x + 2) Middle term 10x + 10x = 20x 25x2 + 20x + 4 = (5x + 2)(5x + 2)

Example: Factor the polynomial 6x2y2 – 2xy2 – 60y2 GCF of 2y2 6x2y2 – 2xy2 – 60y2 = 2y2(3x2 - x - 30) Puzzle pieces for 3x2 x, 3x Puzzle pieces for 30 1, 30 2, 15 3, 10 5, 6 We know the signs 2y2( + )( - ) First guess 2y2(x + 15)(3x - 2) Middle term 45x - 2x = 43x First guess 2y2(x + 3)(3x - 10) Middle term 9x - 10x = -x 6x2y2 – 2xy2 – 60y2 = 2y2(x + 3)(3x - 10)

a) b) c) d)

Example Determine whether each of the following is a perfect-square trinomial. a) x2 + 8x + 16 b) t2  9t  36 c) 25x2 + 4  20x Solution a) x2 + 8x + 16 1. Two terms, x2 and 16, are squares. 2. Neither x2 or 16 is being subtracted. 3. The remaining term, 8x, is 2  x  4, where x and 4 are the square roots of x2 and 16. = (x + 4)2

Example b) t2  9t  36 1. Two terms, t2 and 36, are squares. But 2. Since 36 is being subtracted, t2  9t  36 is not a perfect-square trinomial. c) 25x2 + 4  20x It helps to write it in descending order 25x2  20x + 4 1. Two terms, 25x2 and 4, are squares. 2. There is no minus sign before 25x2 or 4. 3. Twice the product of 5x and 2, is 20x, the opposite of the remaining term, 20x. Thus 25x2  20x + 4 is a perfect-square trinomial. = (5x – 2)2

Factor: 16a2  24ab + 9b2 Solution 16a2  24ab + 9b2 = (4a  3b)2 Example