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Sect. 5.3 Common Factors & Factoring by Grouping Definitions Factor Common Factor of 2 or more terms Factoring a Monomial into two factors Identifying Common Monomial Factors Factoring Out Common Factors Arranging 4 Term Polynomials into 2 Groups Factoring Out Common Binomials 5.3 1 What’s a Factor? product = (factor)(factor)(factor) … (factor) 84 is a product that can be expressed by many different factorizations: 84 = 2(42) or 84 = 7(12) or 84 = 4(7)(3) or 84 = 2(2)(3)(7) Only one example, 84 = 2(2)(3)(7), shows 84 as the product of prime integers. Factoring is the reverse of multiplication. 5.3 2 Factoring Monomials 12x3 also can be expressed in many ways: 12x3 = 12(x3) 12x3 = 4x2(3x) 12x3 = 2x(6x2) Usually, we only look for two factors Your turn – factor these monomials into two factors: 4a = x3 = x(x2) or x2(x) 14y2 = 2(2a) or 4(a) 14(y2) or 14y(y) or 7(2y2) or 7y(2y) or y(14y) or … 43x5 = 43(x5) or 43x(x4) or x3(43x2) or 43x2(x3) or … 5.3 3 Common Factors Sometimes multi-termed polynomials can be factored Looking for common factors in 2 or more terms … is the first step in factoring polynomials Remember a(b + c) = ab + ac (distributive law) Consider that a is a common factor of ab + ac so we can factor ab + ac into a(b + c) For x2 + 3x the only common factor is x , so x2 + 3x = x (? + ?) = x(x + 3) Another example: 4y2 + 6y – 10 The common factor is 2 4y2 + 6y – 10 = 2(? + ? – ?) = 2(2y2 + 3y – 5) 5.3 4 Check by multiplying: 2(2y2) + 2(3y) – 2(5) = 4y2 + 6y – 10 Find the Greatest Common Factor 7a – 21 = 19x3 + 3x = 7(? – ?) = 7(a – 3) x(? + ?) = x(19x2 + 3) 18y3 – 12y2 + 6y = 6y(? – ? + ?) = 6y(3y2 – 2y + 1) 5.3 5 Introduction to Factoring by Grouping: Factoring Out Binomials x2(x + 7) + 3(x + 7) = (x + 7)(? + ?) = (x + 7)(x2 + 3) y3(a + b) – 2(a + b) = (a + b)(? – ?) = (a + b)(y3 – 2) You try: 2x2(x – 1) + 6x(x – 1) + 17(x – 1) = (x – 1)(? + ? – ?) (x – 1)(2x2 + 6x + 17) 5.3 6 Factoring by Grouping 1. 2. For polynomials with 4 terms: Arrange the terms in the polynomial into 2 groups such that each group has a common monomial factor Factor out the common monomials from each group (the binomial factors produced will be either identical or opposites) 3. Factor out the common binomial factor Example: 2c – 2d + cd – d2 2(c – d) + d(c – d) (c – d)(2 + d) 5.3 7 Factor by Grouping 8t3 + 2t2 – 12t – 3 2t2(4t + 1) – 3(4t + 1) (4t + 1)(2t2 – 3) 4x3 – 6x2 – 6x + 9 2x2(2x – 3) – 3(2x – 3) (2x – 3)(2x2 – 3) y4 – 2y3 – 12y – 3 y3(y – 2) – 3(4y + 1) Oops – not factorable via grouping 5.3 8 What Next? Next time: Section 5.4 – Factoring Trinomials 5.3 9