Monday June 30 Factoring Factoring out the GCF
- Slides: 29
Monday, June 30 Factoring
Factoring out the GCF
Greatest Common Factor The greatest common factor (GCF) is the product of what both items have in common. Example: 18 xy , 36 y 2 18 xy = 2 · 3 · x · y 2 36 y = 2 · 3 · y · y GCF = 2 · 3 · y = 18 y
Now you try! Find the greatest common factor of the following: Example 1: 2 2 2 12 a b , 90 a b c 2 GCF = 6 a b Example 2: 2 15 r , 35 s 2 , 70 rs GCF = 5
Factoring - Opposite of distributing - Breaking down a polynomial to what multiplies together to form the polynomial
Example: Factor: 2 12 a + 16 a 1. Factor each term. = 2· 2· 3·a·a + 2· 2·a 2. Pull out the GCF. = 2 · a (3·a + 2· 2) 3. Multiply. = 4 a (3 a + 4) You can check by distributing.
Example: Factor: 2 2 18 cd + 12 c d + 9 cd = 2· 3· 3·c·d·d + 2· 2· 3·c·c·d + 3· 3·c·d = 3 · c · d (2· 3·d + 2· 2·c + 3) = 3 cd (6 d + 4 c + 3)
Now you try! Example 1: 2 15 x + 25 x = 5 x(3 + 5 x) Example 2: 2 2 4 12 xy + 24 xy – 30 x y 3 = 6 xy(2 + 4 y – 5 xy )
Factoring by Grouping
1. Group Factor: terms with ( ). Example: 5 xy – 35 x + 3 y – 21 (5 xy – 35 x) + (3 y – 21) = 5 x (y – 7) + 3 (y – = (5 x + 3) (y – 7) 2. Pull out 7) GCF from each group. 3. Split into factors.
Notes - What is in parentheses MUST be the same!! - Grouping only works if there are 4 terms!!
Now you try! Factor. Example 1: 2 5 y – 15 y + 4 y - 12 = (5 y + 4)(y – 3) Example 2: 2 5 c – 10 c + 2 d – 4 cd = (5 c + 2 d)(1 – 2 c)
2 more important examples: Example 1: 2 xy + 7 x + 2 y + 7 (2 xy + 7 x) + (2 y + 7) = x (2 y + 7) + (2 y 1(2 y++7)7) = (x + 1) (2 y + 7)
Example 2: 15 a – 3 ab – 20 + 4 b (15 a – 3 ab) – (20 + – 4 b) = 3 a (5 – b) – 4 (5 – b) = (3 a – 4) (5 – b) If there is a negative in the middle, you MUST change the sign after it.
Factoring Trinomials
Example 1: 2 Factor: x + 5 x + 6 Look for factors of 6 that ADD to positive 5 (x + 2)(x + 3) 6 1· 6 2· 3
Example 2: 2 Factor: x + 7 x + 12 Look for factors of 12 that ADD to positive 7 (x + 3)(x + 4) 12 1 · 12 2· 6 3· 4
Now you try! 2 Example: x + 6 x + 8 (x + 2)(x + 4) 2 Example: x + 11 x + 10 (x + 1)(x + 10)
To determine the signs: Last sign Positive Middle sign Positive ( + ) Negative ( – ) Negative ( + )( – )
Example 3: 2 Factor: x – 12 x + 27 Look for factors of 27 that ADD to negative 12 (x – 3) (x – 9) 27 1 · 27 3· 9
Example 4: 2 Factor: x + 3 x – 18 Look for factors of 18 that SUBTRACT to positive 3 (x + 6)(x – 3) 18 1 · 18 2· 9 3· 6
Now you try! 2 Example: x – 20 (x + 4)(x – 5) 2 Example: x – 7 x – 18 (x + 2)(x – 9)
Please note! 2 Example: x – 5 x – 6 (x + 1)(x – 6) 2 Example: x – 5 x + 6 (x – 2)(x – 3)
More Factoring Trinomials
Example 1: 2 Factor: 6 x + 17 x + 5 30 1 · 30 22 6 x ++2 x) 2 x + 15 x (6 x (15 x++55) 2 · 15 2 x(3 x + 1) + 5(3 x + 1) 3 · 10 5 · 6 (2 x + 5)(3 x + 1)
Example 2: 2 Factor: 4 x + 24 x + 32 Always check your factors to see if there is anything more that can be factored out.
OR Example 2: 2 Factor: 4 x + 24 x + 32 It is usually faster if you factor out the GCF first. Always check to see if there is anything you can factor out first.
Now you try! 2 Example: 5 x + 27 x + 10 (5 x + 2)(x + 5) 2 Example: 24 x – 22 x + 3 (4 x – 3)(6 x – 1)
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