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Math 90 5.4 "Factoring General Trinomials" Objectives: Factor trinomials of the form ax2 + bx + c where a 6= 1: * * Factor perfect-square trinomials. * Use the grouping method to factor trinomials. Factor Trinomials of the Form ax2+bx+c The FOIL Method: To factor ax2 + bx + c; a 6= 1, using the FOIL method: 1: Factor out the GCF, if one exists. 2: Find two First terms whose product is ax2 : 3: Find two Last terms whose product is c: 4: Look for Outer and Inner products resulting from steps (2) and (3) for which the sum is bx: 5: Check by multiplying. Example 1: (Factoring ax2 + bx + c by using the FOIL method) Factor. a) 3x2 10x 8 b) 24x2 76x + 40 Another method for factoring trinomials of the type ax2 + bx + c; a 6= 1; involves the product, ac, of the leading coe¢ cient a and the last term c. It is called the ac method. Because it uses factoring by grouping, it is also referred to as the grouping method. The ac-Method: To factor a trinomial by grouping 1. Factor out any GCF. 2. Identify a; b; and c, and …nd the key number ac: 3. Find two integers whose product is the key number ac and whose sum is b: 4. Express the middle term, bx; as the sum (or di¤erence) of the numbers from step 3. 5. Enter the two numbers found in step 4 as coe¢ cients of x in the form shown below. 6. Factor the equivalent four-term polynomial by grouping. Page: 1 Notes by Bibiana Lopez Beginning and Intermediate Algebra by Gustafson and Frisk 5.4 Example 2: (Factoring ax2 + bx + c by using the ac-method) Factor the following trinomials. a) 5x2 + 7x + 2 c) 6x2 e) 25a2 b 15x b) 3p2 6 40ab + 10b 4p 4 d) 4 + 11a2 + 6a4 f) 15a2 + 17ab Page: 2 4b2 Notes by Bibiana Lopez Beginning and Intermediate Algebra by Gustafson and Frisk 5.4 Factoring Perfect-Square Trinomials Example 3: (Factoring perfect-square trinomials) Factor the following trinomials. a) 9x2 c) 9 b) 4x2 12x + 4 4x2 4xy y2 d) 16 Page: 3 20x + 25 a2 2ab b2 Notes by Bibiana Lopez