Chapters 8 and 9 Greatest Common Factors & Factoring by Grouping Definitions Factor, Factoring, Prime Polynomial Common Factor of 2 or more terms Factoring a Monomial into two factors Identifying Common Monomial Factors Factoring Out Common Factors Arranging a 4 Term Polynomial into Groups Factoring Out Common Binomials

What’s a Polynomial Factor? product = (factor)(factor)(factor) … (factor) Factoring is the reverse of multiplication. 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. Always try to factor a polynomial into prime polynomials

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 – You try: 4a = 2(2a) or 4(a) x3 = x(x2) or x2(x) 14y2 = 14(y2) or 14y(y) or 7(2y2) or 7y(2y) or y(14y) 43x5 = 43(x5) or 43x(x4) or x3(43x2) or 43x2(x3) or …

Common Factors of Polynomials When a polynomial has 2 or more terms, it may have common factors By definition, a common factor must divide evenly into every term For x2 + 3x the only common factor is x , so x2 + 3x = x·x + x·3 = x (? + ?) = x(x + 3) For 8y2 + 12y – 20 a common factor is 2, so 8y2 + 12y – 20 = 2(? + ? – ?) =2(4y2 + 6y – 10) Check factoring by multiplying: 2(4y2 + 6y – 10) = 8y2 + 12y – 20

The Greatest Common Factor of Polynomials The greatest common factor (or GCF) is the largest monomial that can divide evenly into every term Looking for common factors in 2 or more terms … is always the first step in factoring polynomials Remember a(b + c) = ab + ac (distributive law) Consider that a is a common factor of ab + ac If we find a polynomial has form ab + ac we can factor it into a(b + c) For 3x2 + 3x the greatest common factor is 3x , so 3x2 + 3x = 3x·x + 3x·1 = 3x (? + ?) = 3x(x + 1) Another example: 8y2 + 12y – 20 The GCF is 4 – Divide each term by 4 8y2 + 12y – 20 = 4(? + ? – ?) = 4(2y2 + 3y – 5) Check by multiplying: 4(2y2) + 4(3y) – 4(5) = 8y2 + 12y – 20

Practice: Find the Greatest Common Monomial Factor 7(? – ?) = 7(a – 3) 19y3 + 3y = y(? + ?) = y(19y2 + 3) 8x2 + 14x – 4 = 2(? + ? – ?) = 2(4x2 + 7x – 2) 4y2 + 6y = 2y(? + ?) = 2y(2y + 3)

Find the Greatest Common Factor 18y5 – 12y4 + 6y3 = 6y3(? – ? + ?) = 6y3(3y2 – 2y + 1) 21x2 – 42xy + 28y2 = 7(? – ? + ?) = 7(3x2 – 6xy + 4y2) 22x3 – 110xy2 = 22x(? – ?) = 22x(x2 – 5y2) 7x2 – 11xy + 13y2 = No common factor exists

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)

Practice: Factoring Out Binomials You try: 2x2(x – 1) + 6x(x – 1) – 17(x – 1) = (x – 1)(? + ? – ?) (x – 1)(2x2 + 6x – 17) y2(2y – 5) + x2(2y – 5) = (2y – 5)(? + ?) (2y – 5)(y2 + x2) 5x2(xy + 1) + 6y(xy – 1) = No common factors

Factoring by Grouping Example: 2c – 2d + cd – d2 2(c – d) + d(c – d) For polynomials with 4 terms: Arrange the terms in the polynomial into 2 groups Factor out the common monomials from each group If the binomial factors produced are either identical or opposite, complete the factorization Example: 2c – 2d + cd – d2 2(c – d) + d(c – d) (c – d)(2 + d)

Factor by Grouping 8t3 + 2t2 – 12t – 3 2t2(4t + 1) – 3(4t + 1) (4t + 1)(2t2 – 3)

Factor by Grouping 4x3 – 6x2 – 6x + 9 2x2(2x – 3) – 3(2x – 3) (2x – 3)(2x2 – 3)

Factor by Grouping y4 – 2y3 – 12y – 3 y3(y – 2) – 3(4y – 1) Oops – not factorable via grouping

Grouping Unusual Polynomials x3 – 7x2 + 6x + x2y – 7xy + 6y x(x2 – 7x + 6) + y(x2 – 7x + 6) (x2 – 7x + 6)(x + y) (x – 1)(x – 6)(x + y)

What Next? Section 5.6 – Factoring Trinomials