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Factoring by GCF Remember factor Trees?? Create a factor tree for the following two numbers. 60 24 6 * 10 4 * 6 2*3 * 2*5 2*2 * 2*3 Are there any numbers common to each?? Sure! When we break the numbers down using factor trees, we can see the common factors. In the example to the left, 60 = 2·2·3·5 24 = 2·2·2·3 The common parts are 2, 2, and 3. If we multiply these, we get 2·2·3 = 12. So, 12 is the Greatest Common Factor (GCF). Now, think through the same process using the following sets of numbers: 60 60 =2·2·3·5 75 75 = 3·5·5 45 Break each number down into factors…. 45 = 3·3·5 Now, pick the factors that are in all three… 3·5 = 15 The Greatest Common Factor is 15. We can do the same with monomials!! Find the Greatest Common Factor: 24x 3 y2 = 2·2·2·3·x·x·x·y·y 56x 2 y4 Break each into factors…. = 2·2·2·7·x·x·y·y·y·y Now pick the factors that are in both… 2·2·2·x·x·y·y The GCF = 8x 2 y2 Summarize: What do you notice about the powers of the variables in the GCF of the two monomials above?? The GCF variables have the lowest power for the variable from the monomials! (What is the lowest power of the x for the monomials? x 2 What is the lowest power of the y for the monomials? y 2 These are the powers of the variables in the GCF) When you pull the Greatest Common Factor out of all terms of a polynomial and place the left over parts inside of a parenthesis, it is called Factoring by GCF: 24x 3 y2 + 56x 2 y4 = 8x 2 y2 ( 3x + 7y 2 ) This is called reverse distributive or factoring by GCF Here is another example to factor 60 x3 y 2 z 4 24 xy3 z 3 First find the GCF for 60 x 3 y 2 z 4 and 24 xy3 z 3 ………….. 12 x1 y 2 z 3 is the GCF!!! 60 x 3 y 2 z 4 24 xy3 z 3 factors as 12 xy2 z 3 (5x 2 z 2 y) That means: More examples: 3x + 6xy – 9 x 2 Factors into 3x(1 + 2y - 3 x ) m 2 5m 5 Factors into -( m 2 5m 5 ) ……. Try to leave highest power term + 9x + 6y Factors into 3(3x + 2y) Problems to try: Factor by GCF. 1. 40 x 5 24 x 3 y 2. 18x 3 y 2 45x 2 y 9 xy 3. 32 x 5 48x 3 y 80 xy 2 4. 64 x 4 16 x 3 5. x12 8x 5 12 xy 6. 36 x 5 y 3 z 18x 4 yz 42 xyz 24 xz ANSWERS: 1. 5. 6. 2. 3. 4.