Factoring trinomials ax² + bx +c a = any number besides 1 and 0

Example: 3m² - 24m – 60 Always check for a GCF! GCF: 3 3(m² - 8m – 20) Factor the quotient! 3(m-10)(m + 2)

Practice x³ + 3x² - 4x this is not ax² + bx +c 7x² + 14xy – 21y² 2t⁵ - 14t⁴ + 24t³ X (x² + 3x – 4) X(x +4)(x – 1) 2. 7(x² + 2y – 3y²) 7(x + 3y)(x – 1y) 2t³(t² - 7t + 12) 2t³(t – 4)(t – 3)

Factor by Grouping Example: 5x² + 7x + 2 A = 5 b = 7 c = 2 Remember to always check for a GCF first!!!!! Find 2 factors whose product is (a times c) and the sum is b. 5(2) = 10 Factors of ten that add up to 7 10 and 1 or 5 and 2

Write the middle term, bx, using the factors in step 1. 5x² + 5x + 2x + 2 Factor by grouping 5x(x + 1) + 2 (x + 1) (x + 1)(5x + 2)

Example: 6x² - 5x + 1 6(1) = 6 Factors of 6 that add up to -5 -2 and -3 or -6 and -1 6x² - 3x – 2x + 1 3x(2x – 1) + -1(2x – 1) (2x – 1)(3x – 1)

Practice 2x² - 11x + 12 4x² + 12x + 5 35x² + 4x - 4 2x² - 8x – 3x + 12

Check for a GCF!!!! Remember to always check for a GCF first!!!!! 24x⁴ + 40x³ + 6x² GCF: 2x² 2x²(12x² + 20x + 3) 2x²(12x² + 18x + 2x + 3) 2x²[ 6x(2x + 3) + 1(2x + 3)] 2x²(2x + 3)(6x + 1)

Practice 6xy² + 33xy – 18x 4x³ -9x² - 9x 18x² - 12x + 2 3x(2y² + 11y – 6) 3x(y + 6)(2y – 1) x(4x² - 9x -9) X(4x + 3)(x -3) 3. 2(9x² - 6x + 1) 2(3x -1)(3x-1)

2(3x -1)(3x-1) How can we rewrite this? 2(3x-1)² (3x-1)² is called a perfect square trinomial

Perfect Square Trinomial a² + 2ab + b² = (a+b)² a² - 2ab + b² = (a-b)² Numbers that are Perfect Squares: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121, 12² = 144, etc Check to see if b² is a perfect square and if the coefficient for a² is a perfect square Then check to see if 2(a)(b) = 2ab

x² + 20x + 100 a² = x² b² = 100 a = x b = 10 Check 2ab: 2(x)(10) = 20x (x + 10)² Factors of 100 that add up to 20: 100= 100 and 1; 2 and 50; 4 and 25; 5 and 20; 10 and 10 (x + 10)(x + 10) = (x + 10)²

4m² - 4m + 1 A = 4 b = -4 c = 1 (a)(c) = 4(1) = 4 Factors of 4 that add up to -4 4m² - 2m -2m + 1 2m(2m – 1) + -1(2m – 1) (2m-1)(2m-1) = (2m -1)² Example: 4m² - 4m + 1 a² = 4m² a = 2m b² = 1 b = 1 2ab = 2(2m)(1) = 4m (a-b) ² (2m -1)²

25x² + 25xy + 4y² a² = 25x² a = 5x b² = 4y² b = 2y 2ab = 2(5x)(2y) = 20xy 20xy does not equal 25xy Not a perfect square trinomial Factor by Grouping! 25x² + 25xy + 4y² (a)(c) = 25(4) = 100 Factors: 1 and 100; 2 and 50; 4 and 25; 5 and 20; 10 and 10 25x² + 5xy + 20xy + 4y² 5x(5x + y) + 4y(5x + y) (5x + y)(5x + 4y)