Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 Section 6.1 The Greatest Common Factor and Factoring by Grouping Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

2 Factoring Factoring a polynomial means finding an equivalent expression that is a product. For example, when we take the polynomial and write it as we say that we have factored the polynomial. In factoring, we write a sum as a product.

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4 Greatest Common Factor - GCF The greatest common factor is an expression of the highest degree that divides each term of the polynomial. The variable part of the greatest common factor always contains the smallest power of a variable that appears in all terms of the polynomial. Factoring

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10 Factoring a Monomial from a Polynomial 1.Determine the greatest common factor of all terms in the polynomial. 2.Express each term as the product of the GCF and its other factor. 3.Use the distributive property to factor out the GCF. Factoring

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11 Factor: 7x x The GCF of 7x 2 and 28x is 7x. 7x x = 7x·x +7x·4 = 7x(x + 4) Express each term as the product of the GCF and the its other factor. Factor out the GCF. Determine the GCF. Greatest Common FactorEXAMPLE

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 12 Factor: 7x x The GCF of 7x 2 and 28x is 7x. 7x x = 7x·x +7x·4 = 7x(x + 4) Express each term as the product of the GCF and the its other factor. Factor out the GCF. Determine the GCF. Greatest Common FactorEXAMPLE

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 13 Factor: 5x 4 y − 10x 2 y xy The GCF of 5x 4 y, – 10x 2 y 2, and 25xy is 5xy. 5x 4 y – 10x 2 y xy = 5xy·x 3 − 5xy·2xy + 5xy·5 = 5xy(x 3 – 2xy + 5) Express each term as the product of the GCF and the its other factor. Factor out the GCF. Determine the GCF. Greatest Common FactorEXAMPLE

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 14 Factor: 5x 4 y − 10x 2 y xy The GCF of 5x 4 y, – 10x 2 y 2, and 25xy is 5xy. 5x 4 y – 10x 2 y xy = 5xy·x 3 − 5xy·2xy + 5xy·5 = 5xy(x 3 – 2xy + 5) Express each term as the product of the GCF and the its other factor. Factor out the GCF. Determine the GCF. Greatest Common FactorEXAMPLE

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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 24 Factoring When factoring polynomials, it is preferable to have a first term with a positive coefficient inside parentheses. We can do this with by factoring out −5, the negative of the GCF. Express each term as the product of the negative of the GCF and its other factor. Then use the distributive property to factor out the negative of the GCF.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25 FactoringEXAMPLE SOLUTION Factor: The GCF is 5x. Because the leading coefficient, − 5, is negative, we factor out a common factor with a negative coefficient. We will factor out the negative of the GCF, or − 5x. Factor out the GCF Express each term as the product of the GCF and its other factor

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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 29 Factoring by GroupingEXAMPLE SOLUTION Factor: Identify the common binomial factor in each part of the problem. The GCF, a binomial, is x + y.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 30 Factoring by Grouping Factor out the common binomial factor as follows. Factor out the GCF This step, usually omitted, shows each term as the product of the GCF and its other factor, in that order. CONTINUED

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 31 Factoring by Grouping 1) Group terms that have a common monomial factor. There will usually be two groups. Sometimes the terms must be rearranged. 2) Factor out the common monomial factor from each group. 3) Factor out the remaining common binomial factor (if one exists).

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 32 Factor: 3x 2 + 6x + 4x + 8 by grouping. 3x 2 + 6x + 4x + 8 = (3x 2 + 6x) + (4x + 8) = 3x(x + 2) + 4(x + 2) = (x + 2)(3x + 4) Group terms that have a common monomial factor. Factor out the common monomial factor from each group. Factor out the remaining common binomial factor. Factoring by GroupingEXAMPLE

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 33 Factor: 3x 2 + 6x + 4x + 8 by grouping. 3x 2 + 6x + 4x + 8 = (3x 2 + 6x) + (4x + 8) = 3x(x + 2) + 4(x + 2) = (x + 2)(3x + 4) Group terms that have a common monomial factor. Factor out the common monomial factor from each group. Factor out the remaining common binomial factor. Factoring by GroupingEXAMPLE

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 34 Factor: 5x 2 – xy + 15xy – 3y 2 5x 2 – xy + 15xy – 3y 2 = (5x 2 – xy) + (15xy – 3y 2 ) = x(5x – y) + 3y(5x – y) = (5x – y)(x + 3y) Group terms that have a common monomial factor. Factor out the common monomial factor from each group. Factor out the remaining common binomial factor. Factoring by GroupingEXAMPLE

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 35 Factor: 5x 2 – xy + 15xy – 3y 2 5x 2 – xy + 15xy – 3y 2 = (5x 2 – xy) + (15xy – 3y 2 ) = x(5x – y) + 3y(5x – y) = (5x – y)(x + 3y) Group terms that have a common monomial factor. Factor out the common monomial factor from each group. Factor out the remaining common binomial factor. Factoring by GroupingEXAMPLE

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 36 Factoring by GroupingEXAMPLE SOLUTION Factor: There is no factor other than 1 common to all terms. However, we can group terms that have a common factor: Common factor is : +

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 37 Factoring by Grouping It is sometimes necessary to use a factor with a negative coefficient to obtain a common binomial factor for the two groupings. We now factor the given polynomial as follows: Group terms with common factors Factor out the common factors from the grouped terms Factor out the GCF CONTINUED

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