Factoring Section 4-4

GCF The greatest common factor (GCF) of two or more monomials is the common factor that has the greatest degree and the greatest numerical coefficient. Example: Find the GCF of 20 and 36 Since 20=2∙2∙5 and 36=2∙2∙3∙3 You list all the numbers common to both prime factorization lists, which is 2∙2∙3=9

Examples with Monomials involving Variables Find the GCF of these two monomials: First you have to find the GCF of 9 and 15, which is 3 and you use the least power of each variable that matches in both lists. 3p2

Another Example Find the GCF of these three monomials: Factoring the coefficients, we have: 26=2∙13 39=3∙1 3 78=2∙3∙13 And the GCF of these coefficients is: 13, since it is the only number common to all three lists of factors. Try to find the greatest factors of the variables and click on the sound to see if you’re right! The final answer is: 13p2q2r2

LCM The least common multiple of two or more monomials is the common multiple that has the least degree and the least positive numerical coefficient. Example: Find the LCM of 20 and 35 You still use the prime factorization for this, so the prime factorization of each are: 20=2∙2∙5 and 35=5∙7 You need to list every number that appears in both lists and use the greatest power of each; so: 22∙5∙7=140

More LCM Examples Find the LCM of these two monomials: First, find the LCM of 9 and 15: 9=32 and 15=3∙5 So the LCM of 9 and 15 is: 32∙5=45 So the final answer is: 45p3q

LCM Example Find the LCM of these three monomials: We already found the prime factorization of this since it is the same problem we found the GCF of, so the lists are: 26=2∙13 39=3∙1 3 78=2∙3∙13 The list of each number that occurs in both lists is: 2∙3∙13 and they all of the numbers have just one of those in their lists, so the LCM of the coefficient is 78. Final answer is: 78p3q3r3

Final note: GCFs &LCMs are always positive!!