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Understanding the Greatest Common Factor in Factoring Equations

Learn about the concept of greatest common factor in factoring equations and how to apply it using the distributive law. Practice exercises included.

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Understanding the Greatest Common Factor in Factoring Equations

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  1. Section 3.11 Greatest Common Factors

  2. Questions • What is the “greatest common factor”? • How do we find the greatest common factor? • How can we use the distributive law to “factor out” the greatest common factor?

  3. Definition Factoring is the process of writing a sum or difference as a product.

  4. Definition Factoring is the process of writing a sum or difference as a product. The most simple form of factoring is using the distributive law (backwards).

  5. Definition Factoring is the process of writing a sum or difference as a product. The most simple form of factoring is using the distributive law (backwards). For example: 6x– 3 can be factored as 3(2x– 1).

  6. Definition The greatest common factorof an expression is largest factor that will divide every term of the expression. Numerically: The largest number that divides each coefficient Variables: Each variable that appears in all terms raised to the lowest power that appears in any term

  7. Factoring Out the GCF • Find the GCF of all the terms in the expression. • Write the GCF in front of a set of parentheses. • Fill in the parentheses by determining the terms that are necessary to create the original expression if the distributive law were used. Notes: • The number of terms in the original expression is the same as the number of terms inside the parentheses. • If the terms have no common factor other than 1, the expression cannot be factored using the GCF.

  8. For Example • The GCF of 6x2y + 9xy5 is 3xy so we could factor as 3xy(2x+3y4) • The GCF of 12x2y3– 6x3y + 9x2is 3x2 so we could factor as 3x2(4y3– 2xy + 3)

  9. Exercise 1 Factor out the GCF from each expression. (a) (b) (c)

  10. Exercise 1 Factor out the GCF from each expression. (a) 15x(2x + 5y) (b) 7wz(2x – 4z2 + 1) (c) 4(6 – 2x + 3x3 + 4x5)

  11. Factoring by Grouping When there are 4 or more terms that do not have a common factor to all the terms but do have common factors when considered in groups, factoring by grouping can be used. Consider:

  12. Exercise 2 Use factor by grouping to factor the following: (a) (b)

  13. Exercise 2 Use factor by grouping to factor the following: (a) (x + 5)(x + 9) (b) (y + 11)(x + 10)

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